Geo4.29

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Find the equation to the circle on AB as diameter and hence find the circle passing through A(1,1),B(2,-1),C(3,2)\,

Equation of the circle with AB as diameter is

(x-1)(x-2)+(y-1)(y+1)=0\,

x^{2}-3x+2+y^{2}-1=0\,

x^{2}+y^{2}-3x+1=0\,

Equation of the circle passing through A,B,C is

Let the equation be x^{2}+y^{2}+2gx+2fy+c=0\,

It passes thro'(1,1)

2g+2f+c=-2\,

It passes thro'(2,-1)

4g-2f+c=-5\,

It passes thro'(3,2)

6g+4f+c=-13\,

Solving first two

6g+2c=-7\,

Solving second two,

14g+3c=-23\,

Soving these two equations we get

-10g=25,g={\frac  {-5}{2}},6\cdot {\frac  {-5}{2}}+2c=-7,c=4\,

Now the value of f is

4\cdot {\frac  {-5}{2}}-2f+4=-5,f={\frac  {-1}{2}}\,

Hence the equation is

x^{2}+y^{2}-5x-y+4=0\,

Main Page:Geometry:Circles