Question

The lengths of the tangents from any point on the circle $15x^2+15y^2-48x+64y=0$ to the two circles $5x^2+5y^2-48x+64y+300=0$ and $5x^2+5y^2-24x+32y+75=0$ are in the ratio $2:1$. Is this true or false?

Answer 1

True. Reduce the circles to standard form by making the coefficients of $x^2$ and $y^2$ each unity. If $(p,q$ be any point on $S_1$, then
$p^2+q^2-\frac{48}{15}p+\frac{64}{15}q=0.....\left(1\right)$
$\therefore \frac{t_1^2}{t_2^2}=\frac{p^2+q^2-\frac{48}{5}p+\frac{64}{5}q+60}{p^2+q^2-\frac{24}{5}p+\frac{32}{5}q+15}=\frac{S_2^{\prime }}{S_3^{\prime }}$
Now putting the value of $p^2+q^2$ from $\left(1\right)$ in the above, we get $\frac{t_1}{t_2}=\frac{2}{1}$ etc.