Question

Find the equation of the circle which cuts the circles
$x^2+y^2-9x+14=0$
and $x_2+y^2+15x+14=0$
orthogonally and passes through the point $(2,5)$.

Answer 1

Let the equation of the circle be
$x^2+y^2+2gx+2fy+c=0$.
Since it passes through the point $(2,5)$
$\therefore 4g+10f+c=29.....\left(1\right)$
Also it cuts the two given circles orthogonally and hence applying the condition
$2g_1{g}_2+2f_1{f}_2=c_1+c_2$, we get
$2g(-9/2)+2f\left(0\right)=c+14.....\left(2\right)$
$2g\left(15/2\right)+2f\left(0\right)=c+14.....\left(3\right)$
Subtracting $\left(2\right)$ and $\left(3\right)$, we get
$2g\left(12\right)=0\therefore g=0$.
Hence from $\left(2\right)$.
$c+14=0$ or $c=-14$.
Therefore from $\left(1\right)$ on putting for $g$ and $c$, we get
$10f=-15$ or $f=-3/2$
The required circle is $x^2+y^2-3y-14=0$.