Geo4.125

From Example Problems
Jump to: navigation, search
Find the limiting points of the coaxal system determined by the circles x^{2}+y^{2}-6x-6y+4=0,x^{2}+y^{2}-2x-4y+3=0\,

The equation of the radical axis of the two circles is

(x^{2}+y^{2}-6x-6y+4)-(x^{2}+y^{2}-2x-4y+3)=0\,

-4x-2y+1=0\,

4x+2y-1=0\,

Now the equation of the circle of a coaxal system is

x^{2}+y^{2}-6x-6y+4+k(4x+2y-1)=0\,

x^{2}+y^{2}-2x(3-2k)-2y(3-k)+4-k=0\,

Centre of the circle is \left(3-2k,3-k\right)\,

Radius is {\sqrt  {(3-2k)^{2}+(3-k)^{2}-4+k}}=0\,

Since the circle is limiting point circle,radius is zero.

9+4k^{2}-12k+9+k^{2}-6k-4+k=0\,

5k^{2}-17k+14=0\,

5k^{2}-10k-7k+14=0\,

(5k-7)(k-2)=0\,

k={\frac  {7}{5}},k=2\,

Now the limiting points are

\left({\frac  {1}{5}},{\frac  {8}{5}}\right),(-1,1)\,


Main Page:Geometry:Circles