Geo4.103

Find the equation passing through the origin and cutting the circles $x^2+y^2-4x-6y-3=0,x^2+y^2-16x-6y+4=0\,$ orthogonally.

Let the equation be

$x^2+y^2+2gx+2fy+c=0\,$

This circle passes through the origin,hence

$c=0\,$

The condition for the required circle and the first circle to cut orthogonally is

From the first circle, $g_1=-2,f_1=-3,c_1=-3\,$

$2(g)(-2)+2(f)(-3)=c-3\,$

$-4g-6f=-3\,$

The condition for the required circle and the second circle to cut orthogonally is

From the second circle, $g_2=-8,f_2=-3,c_2=4\,$

$2(g)(-8)+2(f)(-3)=c+4\,$

$-16g-6f=4\,$

Solving the two equations,we get

$-18f=-16,f=\frac{8}{9}\,$

$-16g=20,g=\frac{-7}{12}\,$

Hence the required circle is

$18x^2+18y^2-21x+32y=0\,$