Question

Tangents `OP` and `OQ` are drawn from the origin o to the circle `x^2 + y^2 + 2gx + 2fy+c=0.` Find the equation of the circumcircle of the triangle `OPQ`.

Answer 1

given circle equation
`x^2+2gx+y^2+2fy+c=0`
`x^2+2gx+g^2+y^2+2fy+f^2+c-g^2-f^2=0`
`(x+g)^2+(y+g)^2=(sqrt(g^2+f^2-c))^2`
Let x+g=X and y+f+F
`X^2+Y^2=A^2`
`X^2+Y^2-A^2=0`
`X^2+Y^2-Xx_1-Yy_1=0`
after putting the points
(0+g,0+f)=(g,f)
`(x_1,y_1)=(g,f)`
`X^2+Y^2-Xg-Yf=0`
`(X+g)^2+(Y+f)^2-(x+g)g-(y+f)f=0` `x^2+y^2+2gx+2fy+g^2+f^2-gx-g^2-fy-f^2=0` `x^2+y^2+gx+fy=0`