Proof Problem on Homogeneous Equation Of Second Degree

Question

If the lines represented by the equation $x^2 + y^2= c^2\left(\dfrac{bx+ay}{ab}\right)^2 $ form a right angle, prove that:

$$\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}=\frac{3}{c^2}$$

I don't have enough idea to start.

Answer 1

For two lines, $ax+by+c=0$ and $dx+ey+f=0$ to be right to each other $ad+be$ must be equal to $0$.

In this case, since $(0,0)$ is a solution to the equation, both lines must be of the form $ax+by=0$ and $cx+dy=0$, and the above equation must be of the form $(ax+by)(cx+dy)=acx^2+(ad+bc)xy+bdy^2=0$.

From the condition above, the sum of the coefficients of $x^2$ and $y^2$ must be equal to $0$ for the two lines to be right. One can easily proceed from here.