View source for Geometry

See [[Geometry (encyc)]] in the encyclopedia.

==Basics==
<br>
For these pairs of points, find the midpoint, distance, slope, and equation of the line.<br><br>

1. [[Alg6.1|solution]] <math>(-2,10),(4,9)\,</math>.<br><br>

2. [[Alg6.2|solution]] <math>(12,1),(4,0)\,</math>.<br><br>

3. [[Alg6.3|solution]] <math>(4,13),(0,9)\,</math>.<br><br>

4. [[Alg6.4|solution]] <math>(-5,15),(-5,9)\,</math>.<br><br>

5. [[Alg6.5|solution]] <math>(0,9),(6,-9)\,</math>.<br><br>

6. [[Alg6.6|solution]] Find the value of a if the distance between the points <math>(a,2),(3,4)\,</math> is 2.

7. [[Alg6.7|solution]] Find the relation between x and y,if the point (x,y)is equidistant from <math>(7,-6),(-3,4)\,</math>

8. [[Alg6.8|solution]] Find the area of the triangle formed by <math>(5,2),(-9,-3),(-3,-5)\,</math>

9. [[Alg6.9|solution]] Find the area of the triangle formed by <math>(1,3),(3,-1),(-5,-5)\,</math>

10. [[Alg6.10|solution]] Find the area of the triangle formed by <math>(2,4),(2,6),(2+\sqrt{3},5)\,</math>

11. [[Alg6.11|solution]] Find the value of x if the area of the triangle formed by <math>(10,2),(-3,-4),(x,1)\,</math> is 5 square units

12. [[Alg6.12|solution]] Find the centroid of a triangle whose vertices are formed by <math>(2,7),(3,-1),(-5,6)\,</math>

13. [[Alg6.13|solution]] Find the point which divides the line segment joining (-1,2) and (4,-5) in the ratio 3:2

14. [[Alg6.14|solution]] Find the point which divides the line segment joining (4,5) and (-3,4) in the ratio -6:5

15. [[Alg6.15|solution]] Find the ratio in which the point P(2,1)divides the line segment joining the points A(1,-2) and B(4,7).

16. [[Alg6.16|solution]] Find the ratio in which the X-axis divides the line joining the points (2,4) and (-4,3).

17. [[Alg6.17|solution]] Show that the triangle formed by the points(4,4),(3,5) and (-1,-1) is a right-angled triangle.

18. [[Alg6.18|solution]] Show that the points (-1,7),(3,-5),(4,-8) are collinear.

19. [[Alg6.19|solution]] Find the value of k,if the points (k,2-2k),(-k+1,2k),(-4-k,6-2k) are collinear

20. [[Alg6.20|solution]] Find the equation of the locus of points which are 5 units away from A(4,-3).

21. [[Alg6.21|solution]] Find the equation of the locus of points which are equidistant from <math>A(-3,2),B(0,4)\,</math>

22. [[Alg6.22|solution]] Find the equation of locus of points P such that distance of P from origin is twice the distance of P from <math>A(1,2)\,</math>

23. [[Alg6.23|solution]] Given that the points <math>A(2,3),B(-3,4)\,</math> are points on a triangle, find the locus of P such that the area of the triangle PAB is 8.5 square units.

24. [[Alg6.24|solution]] Find the locus of a point P, the square of whose distance from origin is 4 times its y coordinate.

25. [[Alg6.25|solution]] Find the locus of P if the ratio of the distances from P to <math>(5,-4),(7,6)\,</math> is 2:3.

==Straight Lines-I==

1.[[geo2.1|solution]] Find the equation of the straight line making an angle of <math>120^\circ\,</math> with the X-axis in positive direction and passing through the point <math>(0,-2)\,</math>

2.[[geo2.2|solution]] Find the equation of the straight line which makes intercepts 5 and 6 on the X and Y-axis respectively.

3.[[geo2.3|solution]] Find the equation of the straight line which makes intercepts whose sum is 5 and product is 6.
 
4.[[geo2.4|solution]] Find the equation of the straight line passing through the point <math>(2,3)\,</math> and making intercepts whose sum is zero.

5.[[geo2.5|solution]] Find the slope of the straight line joining <math>(-3,8),(10,5)\,</math>

6.[[geo2.6|solution]] Find the value of x,if the slope of the line joining <math>(2,5),(x,3)\,</math> is 2.

7.[[geo2.7|solution]] Find the equation of the straight line passing through the points <math>(1,1),(2,3)\,</math>

8.[[geo2.8|solution]] Find the equation of a straight line joining the points <math>(at_1^2,2at_1),(at_2^2,2at_2)\,</math>

9.[[geo2.9|solution]] Find the value of y, if the line joining<math>(3,y),(2,7)\,</math> i sparallel to the line joining <math>(-1,4),(0,6)\,</math>

10.[[geo2.10|solution]] Find the equation of the straight line which makes <math>135^\circ\,</math> with X-axis and passing through <math>(3,-2)\,</math>

11.[[geo2.11|solution]] Find the equation of the straight line passing through <math>(-4,5)\,</math> and cutting off equal intercepts on the coordinate axes.

12.[[geo2.12|solution]] Find the equation of the straight line passing through <math>(3,-4)\,</math> and making intercepts in the ratio 2:3.

13.[[geo2.13|solution]] Find the equation of the straight line passing through <math>(4,-3)\,</math> and perpendicular to the line joining <math>(1,1),(2,3)\,</math>.

14.[[geo2.14|solution]] Find the equation of the straight line passing through <math>(4,-3)\,</math> and parallel tothe line joining <math>(6,3),(-4,5)\,</math>.

15.[[geo2.15|solution]] Show that the points <math>(-5,1),(5,5),(10,7)\,</math> are collinear.

16.[[geo2.16|solution]] Show that the points <math>(a,b+c),(b,c+a),(c,a+b)\,</math> are collinear.

17.[[geo2.17|solution]] <math>A(10,4),B(-4,9),C(-2,-1)\,</math> are the vertices of a triangle.Find the equations of i)AB. ii)Median through A. iii)Altitude through B. iv)Perpendicular bisector of side AB. 

18.[[geo2.18|solution]] Find the equation of the straight line whose distance from origin is 4units and the normal from the orgin  to the straight line makes an angle of <math>135^\circ\,</math> with the X-axis in positive direction.

19.[[geo2.19|solution]] Show that the equation of the straight line passing through <math>(x_1,y_1)\,</math> and making an angle of <math>\theta\,</math> with the X-axis in positive direction is <math>\frac{x-x_1}{\cos\theta}=\frac{y-y_1}{\sin \theta}\,</math>. 

20.[[geo2.20|solution]] Find the equation of the straight line in symmetric form having slope <math>\sqrt{3}\,</math> and passing through <math>(2,3)\,</math>.

21.[[geo2.21|solution]]  Find the equation of the straight line in symmetric form having slope <math>\frac{-1}{\sqrt{3}}\,</math> and passing through <math>(-2,0)\,</math>.

22.[[geo2.22|solution]]  Distance of a straight line from the origin is p.The normal on the straight line from the origin makes an angle <math>\alpha\,</math> with the X-axis in positive direction.Find the equations of straight lines whose values are
<math>p=5,\alpha=60^\circ\,</math>

23.[[geo2.23|solution]] Write the various forms of equation of a straight line.  

24.[[geo2.24|Theorm]] If the equations <math>a_1x+b_1y+c=0,a_2+b_2+c_2=0\,</math> represent the same straight line then prove that <math>\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\,</math>

25.[[geo2.25|Theorm]] Change the equation <math>ax+by+c=0\,</math> into normal form.

26.[[geo2.26|Solution]] Transform the equation <math>5x-2y-7=0\,</math> into i)Slope-intercept form ii).Intercept form iii).Normal form.

27.[[geo2.27|Solution]] Transform the equation <math>\sqrt{3}x+y+10=0\,</math> into normal form.

28.[[geo2.28|Theorm]] The ratio in which the straight line <math>L=ax+by+c=0\,</math>divides the line joining the points <math>A(x_1,y_1),B(x_2,y_2)\,</math> is <math>\frac{-L_[x_1,y_1]}{L_[x_2,y_2]}\,</math>.

29.[[geo2.29|Solution]] Find the ratio in which the straight line<math>2x+3y-20=0\,</math> divides the line joining the points <math>(2,3),(7,10)\,</math>

30.[[geo2.30|Solution]] Find the ratio in which the straight line<math>3x-4y-7=0\,</math> divides the line joining the points <math>(2,-7),(-1,3)\,</math>

31.[[geo2.31|Solution]] If <math>(2,1),(-5,7),(-5,-5)\,</math> are mid points of sides of a triangle,find the equations of the sides of a triangle.

32.[[geo2.32|Solution]] Find the point on the straight line <math>3x+y+4=0\,</math> which is equidistant from the points <math>(-5,6),(3,2)\,</math>

33.[[geo2.33|Solution]] If the perpendicular distance of the straight line <math>\frac{x}{a}+\frac{y}{b}=1\,</math>is p, prove that <math>\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}\,</math>.

34.[[geo2.34|Solution]] Find the equation of the straight line passing through the point of intersection of the lines <math>5x-3y-3=0,4x+y-1=0\,</math> and passing through the point <math>(0,2)\,</math>
 
35.[[geo2.35|Solution]] Find the equation of the altitude from A to side BC of triangle ABC formed by <math>A(1,1),B(-3,4),C(2,-5)\,</math>

36.[[geo2.36|Solution]] Find the equations of the medians of the triangle formed by <math>(2,-5),(-3,4),(0,-3)\,</math>

37.[[geo2.37|Solution]] Show that the feet of the perpendicular from <math>(-2,8)\,</math> to the lines <math>x-2y+6=0,x-7y+6=0,x+3y-4=0\,</math> are collinear.

38.[[geo2.38|Solution]] The three straight lines <math>px+qy+r=0,qx+ry+p=0,rx+py+q=0\,</math> are concurrent if <math>p+q+r=0\,</math>

39.[[geo2.39|Solution]] Prove that the perpendicular bisectors of the sides of a triangle are concurrent.

40.[[geo2.40|Solution]] Find the point of intersection of diagonals of the quadrilateral with vertices <math>(1,2),(3,4),(2,1),(-1,-2)\,</math>

41.[[geo2.41|Solution]] Find the value of k if the lines <math>2x+3y=6,kx-9y+4=0,5x+6y=13\,</math> are concurrent.

42.[[geo2.42|Solution]] A variable straight line drawn through the point of intersection of the straight lines <math>\frac{x}{a}+\frac{y}{b}=1\,</math> and <math>\frac{x}{b}+\frac{y}{a}=1\,</math> meets the coordinate axes at A and B. Show that the locus of the midpoint of AB is <math>2(a+b)xy=ab(x+y)\,</math>
<br><br>
43.[[geo2.43|Solution]] Show that the four lines <math>ax\pm by\pm c=0\,</math> form a rhombus whose area is <math>\frac{2c^2}{ab}\,</math>.

44.[[geo2.44|Solution]] Find the circumcenter of the triangle formed by the points <math>(2,1),(1,-2),(-2,3)\,</math>

45.[[geo2.45|Solution]] Two vertices of a triangle are <math>(5,-1),(-2,3)\,</math>. If the orthocenter of the triangle is the origin,find the third vertex.

==Straight Lines-II==

1.[[geo3.1|solution]] Find the equation to the pair of lines passing through the origin and perpendicular to the pair <math>ax^2+2hxy+by^2=0\,</math> is <math>bx^2-2hxy+ay^2=0\,</math>

2.[[geo3.2|solution]] Find the equation to the pair of lines through the origin and forming an equilateral triangle with the line <math>x-2y-1=0\,</math>.Find also the area of the triangle.

3.[[geo3.3|solution]] Find the condition that the lines represented by <math>ax^2+2hxy+by^2=0\,</math> are such that the slope of one line is <math>\lambda\,</math> times that of the other.

4.[[geo3.4|solution]] Find the area of the triangle formed by the lines <math>x^2+4xy+y^2=0,x+y=1\,</math>

5.[[geo3.5|solution]] Find the equation to the two lines represented by the equation <math>x^2-2xy\csc\theta+y^2=0\,</math>

6.[[geo3.6|solution]] Find the centroid of the triangle formed by the lines <math>x^2-5xy+4y^2=0\,</math> and <math>x+2y-6=0\,</math>

7.[[geo3.7|solution]] Show that if one of the lines given by <math>a_1 x^2+2h_1 xy+b_1 y^2=0\,</math> coincides with one of the lines of  <math>a_2 x^2+2h_2 xy+b_2 y^2=0\,</math> then <math>(a_1 b_2-a_2 b_1)^2=4(a_2h_1-a_1h_2)(b_1 h_2-b_2 h_1)\,</math>

8.[[geo3.8|solution]] Show that the lines <math>x^2+16xy-11y^2=0\,</math> form an equilateral triangle with the line <math>2x+y+1=0\,</math> and find its area.

9.[[geo3.9|solution]] The distance of a point <math>P(h,k)\,</math> from a pair of lines passing thro'the origin is d units.Show that the equation of the pair of lines is <math>(xk-hy)^2=d^2(x^2+y^2)\,</math>

10.[[geo3.10|solution]] If <math>ax^2+2hxy+by^2=0\,</math> be two sides of a parallelogram and <math>px+qy=1\,</math> is one diagonal,prove that the other diagonal is <math>y(bp-hq)=x(aq-hp)\,</math>

11.[[geo3.11|solution]] Find the equation to the pair of angle bisectors of the pair of lines <math>(ax+by)^2=3(bx-ay)^2\,</math>

12.[[geo3.12|solution]] If the pair of line <math>ax^2+2pxy-ay^2=0,bx^2+2qxy-by^2=0\,</math> are such that each pair bisects the angle between the other pair,then show that <math>ab+pq=0\,</math>

13.[[geo3.13|solution]] Prove that one of the lines <math>ax^2+2hxy+by^2=0\,</math> will bisect the angle between the coordinate axes if <math>(a+b)^2=4h^2\,</math>

14.[[geo3.14|solution]] Prove that the pair of lines <math>(a-\lambda)x^2+2hxy+(b-\lambda)y^2=0\,</math> is equally inclined with the pair <math>ax^2+2hxy+by^2=0\,</math>

15.[[geo3.15|solution]] Find the bisecting line of the acute angle between the lines<math>3x-4y+2=0,-12x+5y-2=0\,</math>

16.[[geo3.16|solution]] Find the value of k for which the equation <math>12x^2-10xy+2y^2+14x-5y+k=0\,</math> represents two straight lines. Find their point of intersection.

17.[[geo3.17|solution]] Find the value of k for which the equation <math>2x^2+2xy-y^2+kx+6y-9=0\,</math> represents two straight lines. Find their point of intersection.

18.[[geo3.18|solution]] Find the equation to the pair of bisectors of angles between <math>3x^2+4xy-4y^2-11x+2y+6=0\,</math>

19.[[geo3.19|solution]] Find the equation of the lines which pass through the point of intersection of the pair of lines <math>x^2-5xy+4y^2+x+2y-2=0\,</math> and are at right angles to them.

20.[[geo3.20|solution]] If <math>2x^2+3xy+py^2-5x+10y+q=0\,</math> represents a pair of perpendicular lines,find p,q.Find their point of intersection.

21.[[geo3.21|solution]] If <math>ax^2+2hxy+by^2+2gx+2fy+c=0\,</math>represents a pair of lines then show that the square of the distance from the origin to their point of intersection is <math>\frac{c(a+b)-f^2-g^2}{ab-h^2}\,</math>

22.[[geo3.22|solution]] Find k if the equation <math>4x^2+2lxy+y^2+6x+ky=10\,</math> represent a pair of parallel lines.Also find the distance between them.

23.[[geo3.23|solution]] The equation <math>ax^2+2hxy+by^2+2gx+2fy+c=0\,</math> represents a pair of parallel lines. Prove that the equation of the line midway between the two parallel lines is <math>hx+by+f=0\,</math>

24.[[geo3.24|solution]] Show that the pair of lines <math>6x^2+5xy-4y^2+7x+13y-3=0\,</math> form a parallelogram with the pair of lines <math>6x^2+5xy-4y^2=0\,</math>.Find its area.

25.[[geo3.25|solution]] Show that the two pairs of lines <math>12x^2+7xy-12y^2=0,12x^2+7xy-12y^2-x+7y-1=0\,</math> form a square.

26.[[geo3.26|solution]] Show that the lines joining the origin to the points of intersection of two curves <math>ax^2+2hxy+by^2+2gx=0,a_1 x^2+2h_1 xy+by_1^{2}+2g_1 x=0\,</math> will be at right angles to one another if  <math>g(a_1+b_1)=g_1(a+b)\,</math>

27.[[geo3.27|solution]] If the chord <math>x+y=b\,</math> of the curve <math>x^2+y^2-2ax-4a^2=0\,</math> subtends a right angle at the origin ,prove that <math>b(b-a)=4a^2\,</math>

==Circles==

1. i). The equation of a circle whose centre is (a,b) and radius r is <br>
: <math>(x-a)^2+(y-b)^2=r^2\,</math>

ii). The equation of a circle is <math>x^2+y^2+2gx+2fy+c=0\,</math> <br>
: radius is <math>\sqrt{g^2+f^2-c}\,</math>  centre is (-g,-f)

iii). Equation of the circle described on the line segment AB where A=(x1,y1),B=(x2,y2) is <br>
: <math>(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0\,</math>

2.[[geo4.2|solution]] Find the equation to the circle of radius 3 and centre (3,-2).

3.[[geo4.3|solution]] Find the equation of the circle which passes through (-7,1) and has centre (-4,-3).

4.[[geo4.4|solution]] Find the centre and radius of the circle <math>x^2+y^2+6x+8y-96=0\,</math>

5.[[geo4.5|solution]] Find the centre and radius of the circle <math>5x^2+5y^2-2x-3y=0\,</math>

6.[[geo4.6|solution]] If the radius of the circle <math>x^2+y^2-8x+10y+k=0\,</math> is 7,find the value of k.

7.[[geo4.7|solution]] If the equation <math>ax^2+2bxy-2y^2+8x+12y+6=0\,</math> represents a circle,find the values of a and b.

8.[[geo4.8|solution]] Find the position of the point (3,1) with respect to <math>x^2+y^2-2x-4y+3=0\,</math>

9.[[geo4.9|solution]] Find the power of the point (2,-1) with respect to <math>3x^2+3y^2+4x+2y+6=0\,</math>

10.[[geo4.10|solution]] Find the power of the point (a+b,a-b) with respect to <math>x^2+y^2+2bx-3b^2=0\,</math>

11.[[geo4.11|solution]] Find the equation to the point circle with centre (-2,3).

12.[[geo4.12|solution]] Find the equation to the circle passing through (0,0) and concentric with <math>x^2+y^2-2x-3y-4=0\,</math>
<br>
13.[[geo4.13|solution]] One end of the diameter of the circle<math>x^2+y^2-10x-2y+6=0\,</math> is (3,5).Find the other end of the diameter.

14.[[geo4.14|solution]] Find the equation to the circle on the line segment joining the following points as diameter 
i).<math>(5,1),(7,5)\,</math>
ii).<math>(-4,-2),(1,1)\,</math>

15.[[geo4.15|solution]] Find the equation to the circle passing through the points<math>(1,3),(0,-2),(-3,1)\,</math>

16.[[geo4.16|solution]]  Find the equation to the circle passing through the points<math>(a,0),(0,b),(a,b)\,</math>

17.[[geo4.17|solution]]  Show that the points<math>(-6,0),(-2,2),(-2,-8),(1,1)\,</math> are concyclic.

18.[[geo4.18|solution]] Find the circle which passes through (-1,2),(3,-2) and has its centre on the line <math>x-2y=0\,</math>

19.[[geo4.19|solution]] Find the circle which passes through (4,-3),(-1,2) and has its centre on the line <math>3x+4y+1=0\,</math>

20.[[geo4.20|solution]] Find the length of the chord <math>x+2y=5\,</math> of the circle <math>x^2+y^2=9\,</math>

21.
i).The equation of the circumcircle of the triangle formed by the line <math>ax+by+c=0\,</math> with the coordinate axes is <math>ab(x^2+y^2)+c(bx+ay)=0\,</math>

ii).If the two lines<math>a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0\,</math> meet the coordinate axes in four distinct points ,then those points are concyclic if <math>a_1a_2=b_1b_2\,</math>

iii). If the two lines<math>a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0\,</math> meet the coordinate axes in four distinct points ,then the equation to the circle passing through those points is <math>(a_1x+b_1y+c_1)(a_2x+b_2y+c_2)-(a_1b_2+a_2b_1)xy=0\,</math>

iv).If L=0 is a straight line intersecting the circle S=0,then the equation of the circle passing through the points of intersection is <math>S+\lambda L=0\,</math> where L is a parameter.
<br><br>

21.[[geo4.21|solution]] Find the equation to the circumcircle of the traingle formed by the line 7x-3y-2=0 with the coordinate axes.

22.[[geo4.22|solution]] Show that the lines <math>2x-5y+1=0,6y-15x+2=0\,</math> intersect the coordinate axes in concyclic points. Also find the equation of the circle passing through those points.

23.[[geo4.23|solution]] Show that the pair of straight lines <math> ax^2+2hxy+ay^2+2gx+2fy+c=0\,</math>
meet the coordinate axes in concyclic points.Also find the equation of the circle through those cyclic points.

24.[[geo4.24|solution]] Find the equation of the circle which passes through the points of intersection of <math>x^2+y^2-x-y=0\,</math> and  <math>x+y=1\,</math> and also through the point (1,1).

25.[[geo4.25|solution]] Find the equation of the circle which passes through the points of intersection of <math>x^2+y^2+3x-4y+5=0\,</math> and  <math>x-y+2=1\,</math> and also through the point (2,3).

26.[[geo4.26|solution]] Find the equation to the circle passing through the points of intersection of the circle <math>x^2+y^2-8x-2y+7=0\,</math> and the  line <math>4x+12y+1=0\,</math> and which has its centre on y-axis.

27.[[geo4.27|solution]] Find the equation to the circle passing through the points (1,-2),(4,-3) and having the centre on the line <math>3x+4y=7\,</math>

28.[[geo4.28|solution]] Find the equation of the circle which passes through (1,1),(2,2) and whose radius is unity.

29.[[geo4.29|solution]] Find the equation to the circle on AB as diameter and hence find the circle passing through <math>A(1,1),B(2,-1),C(3,2)\,</math>

30.[[geo4.30|solution]] Find the equation to the circle on AB as diameter and hence find the circle passing through <math>A(3,4),B(3,-6),C(-1,2)\,</math>

31.[[geo4.31|solution]] Find the equations of the tangents from the point(0,1) to the circle <math>x^2+y^2-2x-6y+6=0\,</math>

32.[[geo4.32|solution]] Find the locus of the point from which the lengths of the tangents to the circles <math>x^2+y^2+4x+3=0\,</math> and <math>x^2+y^2-6x+5=0\,</math> are in the ratio 2:3. 

33.[[geo4.33|solution]] Find the equations of the tangents to the circle <math>x^2+y^2-4x-6y-12=0\,</math> and parallel to <math>4x-3y=1\,</math>.

34.[[geo4.34|solution]] Find the equations of the tangents to the circle <math>x^2+y^2-2x-4y-4=0\,</math> and parallel to <math>3x-4y-1=0\,</math>.

35.[[geo4.35|solution]] Find the equation of the circle which passes through (1,-2),(3,-4) and touches the X-axis.

36.[[geo4.36|solution]] Prove that the locus of a point tangents from which to the circle <math>x^2+y^2=a^2\,</math> are inclined at an angle alpha is <math>(x^2+y^2-2a^2)^2 (\tan \alpha)^2=4a^2(x^2+y^2-a^2)\,</math>.

37.[[geo4.37|solution]] Find the equations of circles which touch the axis of x at the origin and the line <math>4x-3y+24=0\,</math>

38.[[geo4.38|solution]] Find the locus of point of intersection of two perpendicular tangents to the circle <math>x^2+y^2=a^2\,</math>
 
39.[[geo4.39|solution]] Show that the line x+y+1=0 touches the circle <math>x^2+y^2-3x+7y+14=0\,</math> and find the point of contact.

40.[[geo4.40|solution]] Show that the line 3x=y+13 touches the circle <math>x^2+y^2-4x-6y+3=0\,</math> and find the point of contact.
 
41.[[geo4.41|solution]] Prove that the tangent to the circle <math>x^2+y^2=5\,</math> at (1,-2) also touches the circle <math>x^2+y^2-8x+6y+20=0\,</math> and find the point of contact.

42.[[geo4.42|solution]] Find the equation of the tangent at(1,2) to the circle <math>x^2+y^2+2x-2y-3=0\,</math>. Find also the equation of the tangent parallel to the above tangent.

43.[[geo4.43|solution]] Find the equation to the circle passing through the points of intersection of the lines x+2y-4=0 and the circle <math>x^2+y^2=4\,</math> and touching the line x+2y=5.

44.[[geo4.44|solution]] Find the equation to the circle passing through the points of intersection of the lines x+2y-1=0 and the circle <math>x^2+y^2-2x+1=0\,</math> and touching the line 2x-y+3=0.

45.[[geo4.45|solution]] Find the equation of the circle with centre on the line 2x+y=0 and which touches the lines 4x-3y+10=0 and 4x-3y-30=0.

46.[[geo4.46|solution]] Find the equation of the chord of contact of (4,-1) with respect to the circle <math>2x^2+2y^2=11\,</math>.

47.[[geo4.47|solution]] Find the pole of the line 3x+4y-45=0 with respect to the circle <math>x^2+y^2-6x-8y+5=0\,</math>

48.[[geo4.48|solution]] Show that the lines 2x+3y-12=0 and 3x+2y-2=0 are conjugate lines with respect to the circle <math>x^2+y^2=2\,</math>

49.[[geo4.49|solution]] What is the value of k if (4,k) and (2,3) are conjugate points with respect to the circle <math>x^2+y^2=17\,</math>

50.[[geo4.50|solution]] Find the value of k if the points (4,2) and (8,k) are conjugate with respect to <math>3x^2+3y^2-12x+4y-4=0\,</math>

51.[[geo4.51|solution]] Find the value of k if the lines 2x+3y-4=0 and kx+4y-2=0 are conjugate with respect to <math>x^2+y^2=4\,</math>

52.[[geo4.52|solution]] Show that the poles of tangents of the circle <math>(x-p)^2+y^2=b^2\,</math> with respect to the circle <math>x^2+y^2=a^2\,</math> lie on the curve <math>(px-a^2)^2=b^2(x^2+y^2)\,</math>

53.[[geo4.53|solution]] Find the locus of the point whose polars with respect to the circles <math>x^2+y^2-4x-4y-8=0\,</math> and <math>x^2+y^2-2x+6y-2=0\,</math> are mutually perpendicular.

54.[[geo4.54|solution]] Show that the locus of the poles of the tangents to the circle <math>x^2+y^2=a^2\,</math> with respect to the circle <math>(x+a)^2+y^2=2a^2\,</math> is <math>y^2+4ax=0\,</math>

55.[[geo4.55|solution]] Write down the equation of the chord of the circle <math>x^2+y^2-2x-4y-20=0\,</math> bisected at the point (2,0).

56.[[geo4.56|solution]] Find the equation of the chord of the circles 
i). <math>x^2+y^2=25\,</math> having <math>(1,-1)\,</math> as its midpoint.
ii).<math>x^2+y^2=81\,</math> having <math>(-2,-3)\,</math> as its midpoint.

57.[[geo4.57|solution]] Find the equation of the chord of the circle <math>x^2+y^2+4x-2y-20=0\,</math>having <math>(\frac{1}{2},\frac{-7}{2})\,</math> as its midpoint.

58.[[geo4.58|solution]] Find the midpoint of the chord <math>3x-y-1=0\,</math> with respect to the circle <math>x^2+y^2-4x+6y+1=0\,</math>

59.[[geo4.59|solution]] Find the middle point of the chord of the circle <math>x^2+y^2=9\,</math> intercepted by the line <math>2x+y-5=0\,</math>

60.[[geo4.60|solution]] Find the mid point of the chord of the circle <math>x^2+y^2+4x+6y+2=0\,</math> intercepted by the line <math>3x+5y=13\,</math>

61.[[geo4.61|solution]]Find the equation of the chord of the circle <math>x^2+y^2=15\,</math> having mid point (3,-2). Also find the pole of that chord with respect to the circle.

62.[[geo4.62|solution]]  Find the equation of the chord of the circle <math>x^2+y^2=25\,</math> having mid point (1,2). Also find the pole of that chord with respect to the circle.

63.[[geo4.63|solution]] Find the locus of the midpoints of chords of the circle <math>x^2+y^2=r^2\,</math>,subtending a right angle at the point (a,b).

64.[[geo4.64|solution]] Find the equation of the tangents drawn from the origin to the circle <math>x^2+y^2+10x+10y+40=0\,</math>

65.[[geo4.65|solution]] Find the equation to the pair of tangents drawn from <math>(10,4)\,</math> to the circles<math>x^2+y^2=25\,</math>

66.[[geo4.66|solution]] Show that the pair of tangents drawn from <math>(g,f)\,</math> to the circles<math>x^2+y^2+2gx+2fy+c=0\,</math> are at right angles if <math>g^2+g^2+c=0\,</math>

67.[[geo4.67|solution]] Find the angle between the pair of tangents drawn from (1,3) to the circles <math>x^2+y^2-2x+4y-11=0\,</math>

68.[[geo4.68|solution]] Tangents are drawn to the circle <math>x^2+y^2=a^2\,</math> from a point which always lies on the line <math>lx+my=1\,</math>. Prove that the locus of the mid-point of the chords of contact is <math>x^2+y^2-a^2(lx+my)=0\,</math>.
<br><br>

69.[[geo4.69|solution]] Find the equation of the pair of tangents drawn from the point <math>(4,3)\,</math> to the circle <math>x^2+y^2-2x-4y=0\,</math> and hence find the angle between them.

70.[[geo4.70|solution]] Find the condition that the pair of tangents from the origin to the circle <math>x^2+y^2+2gx+2fy+c=0\,</math> may be at right angles.

71.[[geo4.71|solution]] State whether the following pair of circles intersect or do not intersect or touch each other.
<math>(x-1)^2+(y-3)^2=1\,</math> and <math>(x+3)^2+(y-1)^2=9\,</math>

72.[[geo4.72|solution]] If the polar of the point <math>(x_1,y_1)\,</math> w.r.t the circle <math>x^2+y^2=a^2\,</math> touches the circle <math>(x-a)^2+y^2=a^2\,</math>, show that the point lies on the curve <math>y^2+2ax=a^2\,</math>

73.[[geo4.73|solution]] The polar of P w.r.t the circle <math>x^2+y^2=a^2\,</math> touches the circle <math>(x-f)^2+(y-g)^2=b^2\,</math>.Prove that its locus is given by the equation <math>(fx+gy-a)^2=b^2(x^2+y^2)\,</math>.

74.[[geo4.74|solution]] Find the condition that the two circles <math>x^2+y^2+2a_1 x+2b_1 y=0\,</math> and <math>x^2+y^2+2a_2 x+2b_2 y=0\,</math> touch each other.

75.[[geo4.75|solution]] Find the equation of the common chord of the circles <math>x^2+y^2+4x-12y+14=0\,</math> and<math>x^2+y^2-14x+6y-22=0\,</math>. Find the points of intersection of the circles.Also find the length of the common chord.

76.[[geo4.76|solution]] Find the locus of the poles of the line<math>\frac{x}{a}+\frac{y}{b}=1\,</math> w.r.t the circles which touch the coordinate axes and whose centre lies in the first quadrant.

77.[[geo4.77|solution]] Show that the circle <math>x^2+y^2-4x-6y-12=0,x^2+y^2+6x+18y+26=0\,</math> touch each other and find the point of contact.

78.[[geo4.78|solution]] If the two circles <math>x^2+y^2+2gx+2fy=0,x^2+y^2+2g_1 x+2f_1 y=0\,</math> touch each other,prove that <math>fg_1=f_1 g\,</math>

79.[[geo4.79|solution]] Find the equations of the direct common tangents to the circles <math>x^2+y^2=1,(x-1)^2+(y-3)^2=4\,</math>

80.[[geo4.80|solution]] Find the equation of the pair of direct common tangents to the following circles. 
<math>x^2+y^2=16,x^2+y^2-4x-2y-4=0\,</math>

81.[[geo4.81|solution]] Find the equations to the transverse common tangents of the circles <math>x^2+y^2+4x+2y-4=0,x^2+y^2-4x-2y+4=0\,</math>.

82.[[geo4.82|solution]] Find the equations to the transverse common tangents of the circles <math>x^2+y^2=1,(x-1)^2+(y-3)^2=4\,</math>.

83.[[geo4.83|solution]] Find the equations of common tangents to the circles <math>x^2+y^2-4x-10y+28=0,x^2+y^2+4x-6y+4=0\,</math>.

84.[[geo4.84|solution]] Write down the equation of the common tangent if the two circles <math>x^2+y^2+6x-6y+2=0,x^2+y^2-2x=0\,</math> touch each other.

85.[[geo4.85|solution]] Show that the circles <math>x^2+y^2+2ax+c=0,x^2+y^2+2by+c=0\,</math> touch each other if <math>\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}\,</math>.

86.[[geo4.86|solution]]  Find the length of the common chord of the two circles <math>(x-a)^2+y^2=a^2,x^2+(y-b)^2=b^2\,</math>.

87.[[geo4.87|solution]] Find the length of the common chord of the two circles <math>x^2+y^2+4x-12y+14=0,x^2+y^2-14x+6y-22=0\,</math>

88.[[geo4.88|solution]] Prove that the length of the common chord of the circles <math>(x-a)^2+(y-b)^2=c^2,(x-b)^2+(y-a)^2=c^2\,</math> is <math>\sqrt{[4c^2-2(a-b)^2]}\,</math>. Hence find the condition that the circles may touch.

89.[[geo4.89|solution]] Find the equation to the circle whose diameter is the common chord of two circles <math>x^2+y^2-4x+6y-12=0,x^2+y^2+2x-2y-23=0\,</math>. Hence find the length of the common chord.

90.[[geo4.90|solution]]  Find the equation of the circle described on the common chord of the circles <math>x^2+y^2-4x-6y-12=0,x^2+y^2+6x+4y-12=0\,</math> as diameter.

91.[[geo4.91|solution]]   Find the equation of the circle having the common chord of the circles <math>x^2+y^2+x+5y-2=0,x^2+y^2+2x+5y-4=0\,</math> as diameter.

92.[[geo4.92|solution]]  Show that the length of the common of the two circles <math>x^2+y^2+2gx+c=0\,</math> and <math>x^2+y^2+2fy-c=0\,</math> is <math>2\sqrt{\frac{(g^2-c)(f^2+c)}{g^2+f^2}}\,</math>

93.[[geo4.93|solution]] Find the equation of the circle which passes through the points of intersection of <math>x^2+y^2-2x+1=0,x^2+y^2-5x-6y+4=0\,</math> and touch the line <math>2x-y+3=0\,</math>

94.[[geo4.94|solution]]   Find the equation of the circle which passes through the points of intersection of <math>x^2+y^2=4,x^2+y^2-2x-4y+4=0\,</math> and touch the line <math>x+2y=5\,</math>

95.[[geo4.95|solution]]  Find the equation of the circle whose radius is 5units and which touches the circle <math>x^2+y^2-2x-4y-20=0\,</math> at the point (5,5).

96.[[geo4.96|solution]] Find the equation of the circle of radius <math>2\sqrt{2}\,</math> and passing through the points of intersection of the circles <math>x^2+y^2-2x-4y+4=0,x^2+y^2=4\,</math>.

97.[[geo4.97|solution]] Find the equation of the circle which touches the line <math>x=y\,</math> at the origin and passes through the point <math>(2,1)\,</math>.

98.[[geo4.98|solution]]  Find the angle between the circles <math>x^2+y^2+2x+4y+1=0\,</math> and <math>x^2+y^2+4x+2y+4=0\,</math>

99.[[geo4.99|solution]] Find the acute angle of intersection of the following circles.
<math>(x-2)^2+y^2=2,(x-2)^2+(y-1)^2=1\,</math>

100.[[geo4.100|solution]] If the circles <math>x^2+y^2-8x-6y+21=0\,</math> and <math>x^2+y^2-2y+c=0\,</math> cut each other orthogonally, find the value of c.

101.[[geo4.101|solution]] If the circles <math>x^2+y^2-4x-4y=0\,</math> and <math>x^2+y^2-kx+2y+4=0\,</math> cut each other orthogonally, find the value of k.

102.[[geo4.102|solution]] Find the equation passing through the origin and cutting the circles <math>x^2+y^2-8y+12=0,x^2+y^2-4x-6y-9=0\,</math> orthogonally.

103.[[geo4.103|solution]] Find the equation passing through the origin and cutting the circles <math>x^2+y^2-4x-6y-3=0,x^2+y^2-16x-6y+4=0\,</math> orthogonally.

104.[[geo4.104|solution]] Find the equation passing through the origin and which has its centre on the line<math>x+y=4\,</math> and cuts circle <math>x^2+y^2-4x+2y+4=0\,</math> orthogonally.

105.[[geo4.105|solution]] Find the equation of the circle which cut orthogonally the circles <math>x^2+y^2-6y+1=0,x^2+y^2-4y+1=0\,</math> and touch the line <math>3x+4y+5=0\,</math>

106.[[geo4.106|solution]] Prove that the two circles which pass through the points <math>(0,a),(0,-a)\,</math> and touch the line <math>y=mx+c\,</math> will cut orthogonally if <math>c^2=a^2(2+m^2)\,</math>

107.[[geo4.107|solution]] Find the equation of the circle which cuts orthogonally three circles <math>x^2+y^2+2x+4y+1=0,2x^2+2y^2+6x+8y-3=0\,</math> and <math>x^2+y^2-2x+6y-3=0\,</math>

108.[[geo4.108|solution]] Find the equation of the circle which is orthogonal to each of the circles <math>x^2+y^2-x+y=0,2x^2+2y^2+5x-3y-2=0,x^2+y^2+4x-2y-2=0\,</math>

109.[[geo4.109|solution]] Find the equation of the circle which is orthogonal to each of <math>x^2+y^2-4x-2y+6=0,x^2+y^2-2x+6y=0,x^2+y^2-12x+2y+30=0\,</math>
<br>

110.[[geo4.110|solution]] Find the circle which passes through the points of intersection of the circles <math>x^2+y^2+6x+4y-12=0,x^2+y^2-4x-6y-12=0\,</math> and cuts the circle <math>x^2+y^2-2x+3=0\,</math> orthogonally.

111.[[geo4.111|solution]] Find the equation of the circle which is orthogonal to <math>x^2+y^2+2x+8=0,x^2+y^2-8x+8=0\,</math> and which touches the line <math>x-y+4=0\,</math>

112.[[geo4.112|solution]] Find the equation to the radical axis of the two circles <math>x^2+y^2+10x-4y-1=0,x^2+y^2+5x+y+4=0\,</math>

113.[[geo4.113|solution]] Find the equation of the radical axis of the circles <math>3x^2+3y^2-7x+8y+11=0,x^2+y^2-3x-4y+5=0\,</math>

114.[[geo4.114|solution]] Find the radical centre of the circles <math>x^2+y^2-4x-2y+6=0,x^2+y^2-2x+6y=0,x^2+y^2-12x+2y+30=0\,</math>

115.[[geo4.115|solution]] Find the radical centre of the circles <math>x^2+y^2-2x=0,x^2+y^2+2x=0,x^2+y^2-2y=0\,</math>

116.[[geo4.116|solution]] Find the radical centre of the circles <math>x^2+y^2=9,x^2+y^2-2x-2y-5=0,x^2+y^2+4x+6y-19=0\,</math>

117.[[geo4.117|solution]] Find the equation to the circle which is orthogonal to each of <math>x^2+y^2+2x+17y+4=0,x^2+y^2+7x+6y+11=0,x^2+y^2-x+22y+3=0\,</math>

118.[[geo4.118|solution]] Find the equation to the circle which is orthogonal to each of <math>x^2+y^2+4x+2y+1=0,2x^2+2y^2+8x+6y-3=0,x^2+y^2+6x-2y-3=0\,</math>

119.[[geo4.119|solution]] Find the equation to the circle passing through (1,-1) and belonging to the coaxal system determined by the  circles <math>x^2+y^2-6x-6y+4=0,x^2+y^2-x+2y-3=0\,</math>

120.[[geo4.120|solution]] Find the equation of the circle belonging to the coaxal system determined by the circles <math>x^2+y^2-4x-6y-12=0,x^2+y^2+6x+4y-12=0\,</math> and cuts the circle <math>x^2+y^2-2x+3=0\,</math> orthogonally.

121.[[geo4.121|solution]] Find the equation to the circle touching the line <math>2x-y+3=0\,</math> and belonging to the coaxal system determined by  <math>x^2+y^2-2x+1=0\,</math> and the radical axis <math>x+2y-1=0\,</math>

122.[[geo4.122|solution]] The line <math>x+2y-1=0\,</math> is the radical axis and the circle <math>x^2+y^2=1\,</math> is a member of a coaxal system.Find the circle touching the line <math>x+2y=0\,</math> and belonging to the system.

123.[[geo4.123|solution]] Find the limiting points of the coaxal system determined by the circles <math>x^2+y^2+5x+y+4=0,x^2+y^2+10x-4y-1=0\,</math>

124.[[geo4.124|solution]]  Find the limiting points of the coaxal system determined by the circles <math>x^2+y^2+2x+4y+7=0,x^2+y^2+4x+2y+5=0\,</math>

125.[[geo4.125|solution]]  Find the limiting points of the coaxal system determined by the circles <math>x^2+y^2-6x-6y+4=0,x^2+y^2-2x-4y+3=0\,</math>

126.[[geo4.126|solution]] If (1,2) and(3,1) are the limiting points of a coaxal system of circles find the radical axis.

127.[[geo4.127|solution]] (2,1) is one limiting point of a coaxal system of which the radical axis is <math>x+y+4=0\,</math>.Find the other limiting point.
<br>

128.[[geo4.128|solution]] Find the other limiting point of the coaxal system of which one limiting point is (3,1) and radical axis is<math>x-y+2=0\,</math>

129.[[geo4.129|solution]] Find the equation of the circle which belongs to the coaxal system determined by (0,-3) and (-2,-1) and which is orthogonal to the circle <math>x^2+y^2+2x+6y+1=0\,</math>

130.[[geo4.130|solution]] Find the equation of a circle which passes through the origin and belongs to the coaxal system of which (1,2) (4,3)are the limiting points.

131.[[geo4.131|solution]] Find the equation of the circle belonging to the coaxal system of which the limiting points are <math>(2,-3),(0,-4)\,</math> and which passes through (2,-1)

132.[[geo4.132|solution]] Tangents are drawn parallel to the line <math>y=mx\,</math> to touch the circles of the coaxal system <math>x^2+y^2+2\lambda x+c=0\,</math>. Show that the locus of their points of contact is the curve <math>x^2+2mxy-y^2=c\,</math>

133.[[geo4.133|solution]] Find the equation to the system of circles orthogonal to the coaxal system <math>x^2+y^2+3x+4y-2+k(x+y-7)=0\,</math>

134.[[geo4.134|solution]] Find the coaxal system which is orthogonal to the coaxal sytem <math>x^2+y^2+2x-3y-1+k(x-4y+1)=0\,</math>

135.[[geo4.135|solution]] Show that as k varies the circles<math>x^2+y^2+2ax+2by+2k(ax-by)=0\,</math> form coaxal system.Find the radical axis.
 
136.[[geo4.136|solution]] The origin is a limiting point of a system of coaxal circles of which <math>x^2+y^2+2gx+2fy+c=0\,</math> is a member.Show that the equations of the circles of the orthogonal system are <math>(x^2+y^2)(g+kf)+c(x+ky)=0\,</math> for different values of k.

==Plane==
===The Parabola===
1.[[geo5.1.1|solution]] Write the equation of the parabola whose focus is (1,2) and directrix is <math>x+1=0\,</math>

2.[[geo5.1.2|solution]] Write the equation of the parabola whose focus is (-1,1) and directrix is <math>4x+3y+2=0\,</math>

3.[[geo5.1.3|solution]] Determine the equation of the parabola with vertex at (6,2), its axis parallel to Y-axis and passes through (2,4).

4.[[geo5.1.4|solution]] Find the focus of the parabola i).<math>(y-1)^2=8(x-3\,</math>.
ii).<math>(y-2)^2=8(x-3)\,</math>
iii).<math>y^2-x-2y+2=0\,</math>

5.[[geo5.1.5|solution]] Find the equation of the parabola whose axis is parallel to the Y-axis and which passes through the points <math>(4,5),(-2,11),(-4,21)\,</math>

6.[[geo5.1.6|solution]] Find the equation of the parabola whose axis is parallel to X-axis and which passes through <math>(3,3),(6,5),(6,-3)\,</math>

7.[[geo5.1.7|solution]] Find the equation of the parabola whose axis is parallel to X-axis and passing through the points <math>(-1,3),(-2,1),(1,2)\,</math>

8.[[geo5.1.8|solution]] Find the equation of the parabola whose axis is parallel to Y-axis and passing through <math>(2,5),(1,4),(-1,8)\,</math>

9.[[geo5.1.9|solution]] Find the equation of the parabola whose focus is (3,-4) and directrix is <math>x+y+7=0\,</math>

10.[[geo5.1.10|solution]] Obtain the equation of the parabola whose 
focus is (4,5) and vertex is (3,6)

11.[[geo5.1.11|solution]] Find the vertex,latusrectum,axis,tangent at the vertex,focus and directrix of the parabola <math>x^2+8x+12y+4=0\,</math>

12.[[geo5.1.12|solution]] Find the vertex,latusrectum,axis,tangent at the vertex,focus and directrix of the parabola <math>x^2-2x-4y-3=0\,</math>

13.[[geo5.1.13|solution]] Find the equation of the tangent to the parabola
i)<math>y^2=16x\,</math> inclined at 60 degrees to X-axis.
ii).<math>y^2=8ax\,</math> at <math>(2a,4a)\,</math>

14.[[geo5.1.14|solution]] Find the equation of the normal to the parabola
i).<math>y^2=8x\,</math> at <math>(2,4)\,</math>
ii).<math>y^2=4x\,</math> at <math>(1,2)\,</math>
iii).<math>y^2=4x\,</math> whose slope is 2

15.[[geo5.1.15|solution]] The line <math>2x-y+2=0\,</math> touches the parabola <math>y^2=4px\,</math>.Find p and also the point of contact.

16.[[geo5.1.16|solution]] Find the value of p if the line <math>x+y+2=0\,</math> touches the parabola <math>y^2=px\,</math>

17.[[geo5.1.17|solution]] Find the condition if <math>px+qy+r=0\,</math> is a tangent to <math>y^2=3ax\,</math>

18.[[geo5.1.18|solution]] If the line <math>y=mx+c\,</math> is a tangent to the parabola <math>y^2=4a(x+a)\,</math> prove that the condition is <math>c=am+\frac{a}{m}\,</math>

19.[[geo5.1.19|solution]] Show that the line <math>x-y+2=0\,</math> is a tangent to <math>y^2=8x\,</math>.Find the point of contact.

20.[[geo5.1.20|solution]] Show that the equation of common tangents to the circle <math>x^2+y^2=2a^2\,</math> and the parabola <math>y^2=8ax\,</math> is <math>y=\pm (x+2a)\,</math>

21.[[geo5.1.21|solution]] Find the equations of common tangents to the circle <math>x^2+y^2=8\,</math> and to the parabola <math>y^2=16x\,</math>

22.[[geo5.1.22|solution]] Show that the locus of the point of intersection of perpendicular tangents to the parabola <math>y^2=4ax\,</math> is the directrix <math>x+a=0\,</math>

23.[[geo5.1.23|solution]] Show that the equation of the chord joining the points <math>(x_1,y_1),(x_2,y_2)\,</math> on the parabola <math>y^2=4ax\,</math> is <math>(y-y_1)(y-y_2)=y^2-4ax\,</math>
<br><br>

24.[[geo5.1.24|solution]] Show that the locus of the foot of the perpendicular from the focus 
to the tangent of the parabola <math>y^2=4ax\,</math> is <math>x=0\,</math>,the tangent to the vertex.

25.[[geo5.1.25|solution]] Find the equation to the pair of tangents to the parabola <math>y^2=6x\,</math> which pass through <math>(-1,2)\,</math>

26.[[geo5.1.26|solution]] If a chord of the parabola <math>y^2=4ax\,</math> touches the parabola <math>y^2=4bx\,</math>. Show that the tangents at its extremities meet on the parabola <math>by^2=4a^2x\,</math>

27.[[geo5.1.27|solution]] Find the locus of the midpoints of chords of the parabola <math>y^2=4ax\,</math> which subtend a right angle at the vertex of the parabola.

28.[[geo5.1.28|solution]] Show that the locus of the midpoints of chords of <math>y^2=4ax\,</math> which subtend a constant angle alpha at the vertex is <math>[y^2-2ax+8a^2]^2=16a^2\cot ^{2}\alpha(4ax-y^2)\,</math>.

29.[[geo5.1.29|solution]] Prove that the locus of the midpoints of the focal chords of the parabola <math>y^2=4ax\,</math> is another parabola whose vertex is the focus of <math>y^2=4ax\,</math>.

30.[[geo5.1.30|solution]] Show that the locus of the poles of chords which are normal to the parabola <math>y^2=4ax\,</math> is <math>y^2(x+2a)+4a^3=0\,</math>

31.[[geo5.1.31|solution]] Show that the locus of the poles of the chords of the parabola <math>y^2=4ax\,</math> which subtend a constant angle alpha at the vertex is the curve <math>(x+4a)^2\tan^2\alpha=4(y^2-4ax)\,</math>

32.[[geo5.1.32|solution]] Show that the locus of the poles of chords of the parabola <math>y^2=4ax\,</math> which subtend a right angle at the vertex is<math>x+4a=0\,</math>

33.[[geo5.1.33|solution]] Show that the locus of poles of chords of the parabola <math>y^2=4ax\,</math> which are at a constant distance 'a' from the focus is <math>y^2=4x(2a+x)\,</math>

33.[[geo5.1.33|solution]] The chord of contact of tangents from a point P to the parabola <math>y^2=4ax\,</math> touches the circle <math>x^2+y^2=b^2\,</math>.Prove that the locus of P is<math>4a^2x^2=b^2(y^2+4a^2)\,</math>

34.[[geo5.1.34|solution]] Show that the locus of the midpoints of chords of the prabola<math>y^2=4ax\,</math> which touch the circle <math>x^2+y^2=a^2\,</math> is <math>(y^2-2ax)^2=a^2(y^2+4a^2)\,</math> 

35.[[geo5.1.35|solution]] Show that the locus of poles of chords of the parabola<math>y^2=4ax\,</math> at a constant distance b from the vertex is <math>b^2 y^2+4a^2(b^2-x^2)=0\,</math>
  
36.[[geo5.1.36|solution]] The polar of P w.r.t the parabola <math>y^2=4ax\,</math> touches the circle <math>x^2+y^2=4a^2\,</math>. Find the locus of P.

37.[[geo5.1.37|solution]] Show that the locus of the poles of chords of the parabola <math>y^2=4ax\,</math> which are at constant distance 'd' from the focus is <math>d^2(y^2+4a^2)=4a^2(x+a)^2\,</math>.

38.[[geo5.1.38|solution]] Show that the locus of the midpoints of chords of the parabola <math>y^2=6x\,</math> and which touch the circle <math>x^2+y^2+4x-12=0\,</math> is <math>[y^2-3x-6]^2=16(y^2+9)\,</math>

39.[[geo5.1.39|solution]] A tangent to the parabola <math>y^2+4bx=0\,</math> meets <math>y^2=4ax\,</math> at P and Q. Prove that the locus of the midpoint of PQ is <math>y^2(2a+b)=4a^2x\,</math>

40.[[geo5.1.40|solution]] Prove that the locus of midpoints of chords of constant length 2l of the parabola <math>y^2=4ax\,</math> is <math>(y^2-4ax)(y^2+4a^2)+4a^2 l^2=0\,</math>

41.[[geo5.1.41|solution]] If the normals at the points <math>t_1,t_2\,</math> on the parabola <math>y^2=4ax\,</math> meet on the parabola, prove that <math>t_1 t_2=2\,</math>

42.[[geo5.1.42|solution]] Prove that the locus of the point of intersection of two perpendicular normals to the parabola <math> y^2=4ax\,</math> is the parabola <math>y^2=a(x-3a)\,</math>

43.[[geo5.1.43|solution]] A chord which is normal at "t" to the parabola <math>y^2=4ax\,</math> subtends a right angle at the vertex. Then prove that <math>t=\pm \sqrt{2}\,</math>

44.[[geo5.1.44|solution]] Prove that the circle on a focal radius of a prabola,as diameter touches the tangent at the vertex.

45.[[geo5.1.45|solution]] The line <math>lx+my+na=0\,</math> meets the parabola <math>y^2=4ax\,</math> at P,Q. The lines joining P and Q to the focus meet the parabola in M,N.Show that the equation to MN is <math>nx-my+la=0\,</math>

46.[[geo5.1.46|solution]] Show that the locus of the point,two of the normals from which to the parabola <math>y^2=4ax\,</math> are coincident is <math>27ay^2=4(x-2a)^3\,</math>

47.[[geo5.1.47|solution]] From the points of <math>2x-3y+4=0\,</math> tangents are drawn to <math>y^2=4ax\,</math>. Show that the chords of contact pass through a fixed point.

48.[[geo5.1.48|solution]] P is a point on the line <math>lx+my+n=0\,</math>.The polar of P w.r.t the parabola <math>y^2=4ax\,</math> meets the curve in Q and R.Show that the locus of the midpoint of QR is <math>l(y^2-4ax)+2a(lx+my+n)=0\,</math>

49.[[geo5.1.49|solution]] Show that the tangent at one extremity of a focal chord of a parabola is parallel to the normal at the other extremity.

50.[[geo5.1.50|solution]] Prove that the length of the chord of contact of tangents drawn from <math>(x_1,y_1)\,</math> to the parabola <math>y^2=4ax\,</math> is  <math>\frac{\sqrt{(y_1^{2}-4ax_1)}\sqrt{(y_1^{2}+4a^2)}}{a}\,</math>.

===The Ellipse===
1.[[geo5.2.1|solution]] Find the eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse <math>9x^2+16y^2=144\,</math>

2.[[geo5.2.2|solution]] Find the lengths of major axis, minor axis, latus rectum, eccentricity, centre, foci, equations of directrices of the ellipse  <math>\frac{x^2}{64}+\frac{y^2}{36}=1\,</math>

3.[[geo5.2.3|solution]]  Find the eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse <math>4x^2+y^2-8x+2y+1=0\,</math>

4.[[geo5.2.4|solution]] Find the lengths of major axis, minor axis, latus rectum, eccentricity, centre, foci, equations of directrices of the ellipse  <math>x^2+2y^2-4x+12y+14=0\,</math>

5.[[geo5.2.5|solution]] Find the lengths of major axis, minor axis, latus rectum, eccentricity, centre, foci, equations of directrices of the ellipse  <math>3x^2+y^2-6x-2y-5=0\,</math>

6.[[geo5.2.6|solution]] If the two ends of major axis of an ellipse are (5,0),(-5,0). Find the equation of ellipse if its focus lies on the line <math>3x-5y-9=0\,</math>

7.[[geo5.2.7|solution]] Find the equation of the ellipse in the usual form,if it passes through the points (-2,2) and (3,1).(axis are along the coordinate axes and centre at the origin).

8.[[geo5.2.8|solution]] Find the equation of the ellipse with a focus at(1,-1),e=2/3 and directrix is <math>x+y+2=0\,</math>

9.[[geo5.2.9|solution]] Find the eccentricity of the ellipse ,if its length of the latus rectum is equal to half of its major axis.

10.[[geo5.2.10|solution]] Find the equation of the ellipse referred to its major axis and minor axis as the axes of coordinates a and y axes respectively with latus rectum of length 4 and distance between foci <math>4\sqrt{2}\,</math>

11.[[geo5.2.11|solution]] Find the equation of ellipse with length of latus rectum 15/2 and distance between foci 2.

12.[[geo5.2.12|solution]]Show that the condition for a straight line <math>y=mx+c\,</math> be a tangent to the ellipse<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> is <math>c^2=a^2 m^2+b^2\,</math>

13.[[geo5.2.13|solution]] If the length of the latus rectum is equal to half of its minor axis of an ellipse in the standard form,then find the eccentricity of the ellipse.

14.[[geo5.2.14|solution]] Find the equations of tangent and normal to the ellipse <math>2x^2+3y^2=11\,</math> at the point whose ordinate is 1.

15.[[geo5.2.15|solution]] Find the equation of the tangent and normal to the ellipse <math>x^2+2y^2-4x+12y+14=0\,</math> at (2,-1).

16.[[geo5.2.16|solution]] Find the equation of tangents to the ellipse <math>2x^2+y^2=8\,</math> which is parallel to <math>x-2y-4=0\,</math>.

17.[[geo5.2.17|solution]] Find the equations to the tangents to the ellipse <math>x^2+2y^2=3\,</math> drawn from the point (1,2).

18.[[geo5.2.18|solution]] Show that the foot of the perpendicular drawn from the centre on any tangent to the ellipse lies on the curve <math>(x^2+y^2)^2=a^2 x^2+b^2 y^2\,</math>

19.[[geo5.2.19|solution]] If the normal at one end of a latus rectum of the ellipse <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> passes through one end of the minor axis,then show that <math>e^4+e^2=1\,</math>

20.[[geo5.2.20|solution]] Show that the points of intersection of the perpendicular tangents to an ellipse lies on a circle.

21.[[geo5.2.21|solution]] Find the equation of the tangents to the ellipse <math>2x^2+y^2=8\,</math> is perpendicular to <math>x+y+2=0\,</math>

22.[[geo5.2.22|solution]] Find the coordiantes of the points on the ellipse <math>x^2+3y^2=37 \,</math> at which the normal is parallel to the line <math>6x-5y=2\,</math>

23.[[geo5.2.23|solution]] Prove that the sum of the squares of the perpendiculars on any tangent of the ellipse <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math>(a > b) from two points on the minor axis,each at a distance of <math>\sqrt{a^2-b^2}\,</math> from the centre is <math>2a^2\,</math>
<br><br>


24.[[geo5.2.24|solution]] Find the locus of the point of intersection of the two tangents to the ellipse <math>b^2 x^2+a^2 y^2=a^2 b^2\,</math>,which include an angle theta.

25.[[geo5.2.25|solution]] Find the pole of the line <math>x-y+2=0\,</math> with w.r.t the ellipse <math>x^2+2y^2-4x+12y+14=0\,</math>

26.[[geo5.2.26|solution]] Find the pole of the line <math>21x-16y-12=0\,</math> w.r.t the ellipse <math>3x^2+4y^2=12\,</math>

27.[[geo5.2.27|solution]] Find the pole of the line <math>3x-5y-9=0\,</math> w.r.t the ellipse <math>4x^2+8y^2-16x+15=0\,</math>

28.[[geo5.2.28|solution]] Find the equation of a straight line through the point (2,1) and conjugate to the straight line <math>9x+2y=1\,</math> w.r.t the ellipse <math>3x^2+2y^2=1\,</math>

29.[[geo5.2.29|solution]] Show that the two lines <math>9x+2y=1,3x+2y=11\,</math> are conjugate w.r.t the ellipse <math>3x^2+2y^2=1\,</math>

30.[[geo5.2.30|solution]] Find the value of k,if the lines<math>x+y+k=0,3x-2y-7=0\,</math> are conjugate w.r.t the ellipse <math>x^2+3y^2=9\,</math>

31.[[geo5.2.31|solution]] Find the value of k if <math>(1,2),(k,-1)\,</math> are conjugate w.r.t the ellipse <math>2x^2+3y^2=6\,</math>

32.[[geo5.2.32|solution]] Show that the poles of the tangents of <math>y^2=4kx(k > 0)\,</math> w.r.t the ellipse <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> lie on a parabola.

33.[[geo5.2.33|solution]] Show that the poles of normal chords of the ellipse<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> lie on the curve <math>\frac{a^6}{x^2}+\frac{b^6}{y^2}=(a^2-b^2)^2\,</math>

34.[[geo5.2.34|solution]] Show that the poles of the tangents to the circle <math>x^2+y^2=a^2+b^2\,</math> w.r.t the ellipse <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> lies on <math>\frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{a^2+b^2}\,</math>

35.[[geo5.2.35|solution]] Show that the poles of the tangents to the auxiliary circle w.r.t the ellipse <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> is the curve <math>\frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{a^2}\,</math>

36.[[geo5.2.36|solution]] Prove that the product of the perpendicular from the foci on any tangent to the ellipse <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> is equal to <math>b^2\,</math>
 
37.[[geo5.2.37|solution]] Find the equation of the pair of tangents to the ellipse <math>x^2+3y^2=3\,</math> from the point <math>(2,-1)\,</math>.

38.[[geo5.2.38|solution]] A chord PQ of an ellipse subtends a right angle at the centre of the ellipse <math>S=0\,</math>.Show that the point of intersection of tangents at P and Q lies on the ellipse<math>\frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{a^2}+\frac{1}{b^2}\,</math>

39.[[geo5.2.39|solution]] Prove that the pair of tangents drawn to <math>9x^2+16y^2=144\,</math> are perpendicular to eachother.

40.[[geo5.2.40|solution]] Show that the equation of the auxiliary circle of the ellipse <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,</math> is <math>x^2+y^2=a^2\,</math>
 
41.[[geo5.2.41|solution]] Tangents at right angles are drawn to the ellipse<math>S=0\,</math>.Show that the locus of the midpoints of chords of contact is the curve <math>[\frac{x^2}{a^2}+\frac{y^2}{b^2}]^2=\frac{x^2+y^2}{a^2+b^2}\,</math>

42.[[geo5.2.42|solution]] Find the locus of the midpoints of chords of an ellipse,whose poles lie on the auxiliary circle.

43.[[geo5.2.43|solution]] P is a point on the ellipse <math>S=0\,</math> and Q is its corresponding point on the auxiliary circle. Prove that the locus of the point of intersection of the normals at P and Q is the circle given by <math>x^2+y^2=(a+b)^2\,</math>

44.[[geo5.2.44|solution]] Find the equation of the ellipse whose vertices are <math>(-4,1),(6,1)\,</math> and whose focus lies on the line <math>x-2y=2\,</math>

45.[[geo5.2.45|solution]] Find the equation of the ellipse whose vertices are <math>(-4,3),(8,3)\,</math> and whose eccentricity is 5/6.

46.[[geo5.2.46|solution]] Find the point of contact of the tangent line <math>4x+y-7=0\,</math> to the ellipse <math>x^2+3y^2=3\,</math>

47.[[geo5.2.47|solution]] Find the value of k if the line <math>x+ky-5=0\,</math> is a tangent to the ellipse <math>4x^2+9y^2=20\,</math>

48.[[geo5.2.48|solution]] Find the value of k and hence the point of contact of the tangent line <math>4x+y+k=0\,</math> with the ellipse <math>x^2+3y^2=3\,</math>

49.[[geo5.2.49|solution]] Find the equations to the tangents to the ellipse <math>4x^2+3y^2=5\,</math> which are parallel to <math>3x-y+7=0\,</math>

50.[[geo5.2.50|solution]] Show that the locus of poles of chords of ellipse <math>S=0\,</math> which touch the parabola <math>y^2=4px\,</math> is <math>pa^2 y^2+b^4 x=0\,</math>

===Hyperbola===
1.[[geo5.3.1|solution]] Write down the equation to the hyperbola whose focus is <math>(4,0)\,</math>,directrix is the line <math>4x-9=0\,</math> and eccentricity is 4/3.

2.[[geo5.3.2|solution]] Find the equation to the hyperbola whose focus is <math>(1,2)\,</math> eccentricity <math>\sqrt{3}\,</math> and directrix is <math>2x+y-1=0\,</math>

3.[[geo5.3.3|solution]] What is the equation to the hyperbola if the latusrectum is 9/2 and eccentricity is 5/4.

4.[[geo5.3.4|solution]] Obtain the equation of the hyperbola in standard form whose latusrectum is 4 and eccentricity is 3.

5.[[geo5.3.5|solution]] Determine the equation to the hyperbola whose centre is (0,0),distance between the foci is 18 and that between the directrices is 8.

6.[[geo5.3.6|solution]] A hyperbola has one focus at the origin and its eccentricity is <math>\sqrt{2}\,</math>.One of its directrices is<math>x+y+1=0\,</math>.Find the equation of the hyperbola.

7.[[geo5.3.7|solution]] Find the centre,eccentricity,length of latusrectum,foci,vertices and equations to the directrices of the hyperbola <math>\frac{(y-4)^2}{9}-\frac{(x-5)^2}{16}=1\,</math>

8.[[geo5.3.8|solution]] Find the centre,eccentricity,length of latusrectum,foci,vertices and equations to the directrices of the hyperbola <math>9x^2-16y^2-18x-32y-151=0\,</math>

9.[[geo5.3.9|solution]] What are the coordinates of the foci of the hyperbola <math>\frac{(x+1)^2}{25}-\frac{(y+2)^2}{16}=1\,</math>

10.[[geo5.3.10|solution]] Write down the equations of the directrices of the hyperbola <math>\frac{x^2}{144}-\frac{y^2}{36}=1\,</math>

11.[[geo5.3.11|solution]] Show that the ellipse <math>7x^2+16y^2=112\,</math> and the hyperbola <math>\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}\,</math> have the same foci.

12.[[geo5.3.12|solution]] Find the equations to the tangents to the hyperbola <math>3x^2-y^2=3\,</math> which are perpendicular to <math>x+3y=2\,</math>

13.[[geo5.3.13|solution]] Find the equations of the tangents to the hyperbola <math>3x^2-4y^2=12\,</math> which make equal intercepts on the axes.

14.[[geo5.3.14|solution]] Find the value of k if the line <math>3x-y=k\,</math> is a tangent to <math>3x^2-y^2=3\,</math>

15.[[geo5.3.15|solution]] Find the equations of tangents to the hyperbola <math>4x^2-3y^2=24\,</math> which make an angle of 60 degrees with X-axis.  

16.[[geo5.3.16|solution]] Prove that the line <math>21x+5y-116=0\,</math> touches the hyperbola <math>7x^2-5y^2=232\,</math> and find the point of contact.

17.[[geo5.3.17|solution]] Show that the line <math>x\cos\alpha+y\sin\alpha=p\,</math> touches the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> if <math>a^2 \cos^2 \alpha-b^2 \sin^2 \alpha=p^2\,</math>

18.[[geo5.3.18|solution]] Find the equations of the tangents to the hyperbola <math>9x^2-16y^2=1\,</math> drawn parallel to to the line <math>9x+8y=10\,</math>

19.[[geo5.3.19|solution]] Find the equation of the normal at (1,0)on the hyperbola <math>x^2-4y^2=1\,</math>

20.[[geo5.3.20|solution]] Show that the locus of poles w.r.t hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> of tangents to the parabola <math>y^2=4ax\,</math> is <math>a^3 y^2+b^4 x=0\,</math>

21.[[geo5.3.21|solution]] Show that the locus of the pole of any tangent to the circle <math>x^2+y^2=a^2\,</math> w.r.t the hyperbola <math>x^2-y^2=a^2\,</math> is the circle itself.
<br><br>


22.[[geo5.3.22|solution]] The polar of any point on the ellipse <math>S=0\,</math> w.r.t the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> will touch the ellipse.

23.[[geo5.3.23|solution]] If the polar of a point w.r.t ellipse <math>S=0\,</math> touch the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math>,then show that the locus of point is is the hyperbola.

24.[[geo5.3.24|solution]] Prove that the locus of points the polars of which w.r.t the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> touch the circle <math>x^2+y^2=c^2\,</math> i f <math>\frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{c^2}\,</math>

25.[[geo5.3.25|solution]] Show that the locus of the poles w.r.t the parabola <math>y^2=4ax\,</math> of tangents to the hyperbola <math>4x^2-3y^2=a^2\,</math> is <math>12x^2+y^2=3a^2\,</math>

26.[[geo5.3.26|solution]] Find the line passing through the point (-2,1) and conjugate to the line <math>8x+3y-4=0\,</math> w.r.t <math>2x^2-y^2=1\,</math>

27.[[geo5.3.27|solution]] Find the equation to the line passing through (1,2) and conjugate to the line <math>3x+2y+6=0\,</math> w.r.t hyperbola <math>3x^2-4y^2=12\,</math>

28.[[geo5.3.28|solution]] Show that the locus of poles w.r.t parabola <math>y^2=4ax\,</math> of the tangents to the hyperbola <math>x^2-y^2=a^2\,</math> is the ellipse <math>4x^2+y^2=4a^2\,</math>

29.[[geo5.3.29|solution]] Show that the locus of the foot of perpendicular from the centre of the hyperbola <math>\frac{x_2}{a^2}-\frac{y^2}{b^2}=1\,</math> on a variable tangent is <math>(x^2+y^2)^2=a^2 x^2-b^2 y^2\,</math>

30.[[geo5.3.30|solution]] Tangents to the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> make angles <math>\alpha,\beta\,</math> with the transverse axis. Find the locus of their point of intersection if <math>\tan\alpha+\tan\beta=k\,</math>
<br>

31.[[geo5.3.31|solution]] Tangents drawn from <math>(\alpha,\beta)\,</math> to the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> make an angle <math>\theta_1,\theta_2\,</math> with the x-axis.If <math>\tan\theta_1 \tan\theta_2=1\,</math> prove that<math>\alpha^2-\beta^2=a^2+b^2\,</math>

32.[[geo5.3.32|solution]] Find the equation of the chord of the hyperbola <math>4x^2-y^2=4\,</math> which is bisected at the point <math>(2,-3)\,</math>

33.[[geo5.3.33|solution]] Find the equation to the hyperbola whose asymptotes are <math>3x=\pm5y\,</math> and vertices are <math>(\pm5,0)\,</math>

34.[[geo5.3.34|solution]] The asymptotes of the hyperbola are parallel to the lines <math>2x+3y
=0,3x+2y=0\,</math>. Its centre is at <math>(1,2)\,</math> and passes thro' the point <math>(5,3)\,</math>.Find its equation.

35.[[geo5.3.35|solution]]  Find the equation of the hyperbola whose asymptotes are <math>7x+5y-3=0,2x+4y+1=0\,</math> and passing thro' <math>(-2,1)\,</math>

36.[[geo5.3.36|solution]]  Find the equation of the hyperbola whose asymptotes are <math>2x+3y-5=0,5x+3y-8=0\,</math> and passing thro' <math>(1,-1)\,</math>

37.[[geo5.3.37|solution]] Find the locus of midpoints of the chords of the parabola <math>\frac{x^2}{9}-\frac{y^2}{4}=1\,</math> which are parallel to <math>3x+8y-4=0\,</math>

38.[[geo5.3.38|solution]] Show that the locus of midpoints of the chord of the hyperbola<math>x^2-y^2=a^2\,</math>,which touch the parabola <math>y^2=4ax\,</math> is <math>y^2(x-a)=x^3\,</math>

39.[[geo5.3.39|solution]]  Show that the locus of midpoints of the chord of the hyperbola<math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math>,which pass through the focus <math>(ae,0)\,</math> is <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{ex}{a}\,</math>

40.[[geo5.3.40|solution]] P is any point on the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> whose vertex is A(a,0).Show that the locus of the middle point of AP is <math>\frac{(2x-a)^2}{a^2}-\frac{4y^2}{b^2}=1\,</math>

41.[[geo5.3.41|solution]] A tangent of the auxiliary circle of the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> intersects it in P and Q.Find the locus of midpoint of PQ.

42.[[geo5.3.42|solution]] Find the locus of midpoints of the chords of hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> drawn parallel to the line <math>y=mx\,</math>

43.[[geo5.3.43|solution]] Find the asymptotes of the hyperbola <math>2x^2+5xy+2y^2-11x-7y-4=0\,</math>

44.[[geo5.3.44|solution]]  Find the asymptotes of the hyperbola <math>2x^2-xy-y^2+2x-2y+2=0\,</math>

45.[[geo5.3.45|solution]] If e1,e2 are the eccentricities of a hyperbola and its conjugate,prove that <math>\frac{1}{e_1^{2}}+\frac{1}{e_2^{2}}=1\,</math>

46.[[geo5.3.46|solution]] If a tangent at a point P to a hyperbola meets the asymptotes in Q and R show that P is the midpoint of QR.

47.[[geo5.3.47|solution]] Show that the points of intersection of the asymptotes of hyperbola with its directrices lie on the auxiliary circle.

48.[[geo5.3.48|solution]] Show that the midpoints of normal chords of <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> is <math>[\frac{a^6}{x^2}-\frac{b^6}{y^2}][\frac{x^2}{a^2}-\frac{y^2}{b^2}]^2=(a^2+b^2)^2\,</math>

49.[[geo5.3.49|solution]] Show that the locus of the foot of the perpendicular drawn from the centre of the hyperbola <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> on any normal to it is <math>(a^2 y^2-b^2 x^2)(x^2+y^2)^2=(a^2+b^2)x^2 y^2\,</math>

50.[[geo5.3.50|solution]] Prove that the product of the perpendiculars from any point on the hyperbola<math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,</math> to its asymptotes is constant.

51.[[geo5.3.51|solution]]Show that the portion of any tangent to the hyperbola,intercepted between the asymptotes is bisected at the point of contact.

===Polar Coordinates===
1.[[geo5.4.1|solution]] What are the polar coordinates of <math>(-1,-1)\,</math>

2.[[geo5.4.2|solution]] What are the cartesian coordinates of the points
  i). <math>(3,\frac{\pi}{6})\,</math>
  ii). <math>(4,\frac{\pi}{3})\,</math>

3.[[geo5.4.3|solution]] Determine the lengths of the sides of the triangle whose vertices are <math>(-3,\frac{\pi}{6}),(5,\frac{5\pi}{6}),(7,\frac{7\pi}{6})\,</math>

4.[[geo5.4.4|solution]] Find the distance between the points below

i).  <math>(-1,\frac{\pi}{8}),(\sqrt{2},\frac{3\pi}{8})\,</math>

ii). <math>(-3,45^\circ),(7,105^\circ)\,</math>

5.[[geo5.4.5|solution]] Find the area of the triangle formed by the points <math>(-3,-30^\circ),(5,150^\circ),(7,120^\circ)\,</math>

6.[[geo5.4.6|solution]] Find the area of the triangle formed by the points <math>(1,\frac{\pi}{6}),(2,\frac{\pi}{3}),(3,\frac{\pi}{2})\,</math>

7.[[geo5.4.7|solution]] Prove that the points <math>(\frac{3}{2},0),(-\sqrt{2},45^\circ),(-\frac{3}{5},90^\circ)\,</math> are collinear.

8.[[geo5.4.8|solution]] Find the equation of the line joining the points <math>(3,\frac{3\pi}{4}),(2,\frac{\pi}{4})\,</math>

9.[[geo5.4.9|solution]] Find the equation of the line joining the points <math>(2,\frac{\pi}{6}),(3,\frac{\pi}{3})\,</math>

10.[[geo5.4.10|solution]]  Find the equation of the line passing thro'the point <math>(2,\frac{\pi}{6})\,</math> parallel and perpendicular to the line <math>\frac{3}{r}=4\cos\theta-3\sin\theta\,</math>

11.[[geo5.4.11|solution]] Find the equation of the line passing thro' <math>(2,\frac{\pi}{3})\,</math> and parallel to  <math>r(4\cos\theta+\sqrt{3}\sin\theta)=2\,</math>

12.[[geo5.4.12|solution]] Find the equation of the line passing thro' <math>(-1,\frac{\pi}{2})\,</math> and parallel to <math>4=r(2\cos\theta+\sqrt{3}\sin\theta)\,</math>   

13.[[geo5.4.13|solution]] Find the length of the perpendicular from the origin on the line <math>\frac{8}{r}=\sqrt{3}\cos\theta+\sin\theta\,</math>.Also determine the angle made by the perpendicular with the intial line.

14.[[geo5.4.14|solution]] Find the perpendicular distance from the origin to <math>6=r(\cos\theta+\sqrt{3}\sin\theta)\,</math>

    
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