Question

What is the general equation of a circle passing through two given points $(x_1,y_1)$ and $(x_2,y_2)$? If $S_1,S_2,S_3$ be three members of this family and $t_2,t_3$ be the tangents from any point on $S_1$ to circles $S_2$ and $S_3$, then show that $t_2/t_3$ is constant.

Answer 1

$S+\lambda P=0$ is the required equation where $S=0$ is a circle on given points as diameters and $P=0$ is the line joining these two points.
Let $S_1,S_2,S_3$ be the three members of this family corresponding to ${\lambda }_1,{\lambda }_2,{\lambda }_3$.
Let $P(h,k)$ be any point on $S_1$
$\therefore S^{\prime }+{\lambda }_1{P}^{\prime }=0.....\left(1\right)$
Now $\frac{t_2^2}{t_3^2}=\frac{S^{\prime }+{\lambda }_2{P}^{\prime }}{S^{\prime }+{\lambda }_3{P}^{\prime }}=\frac{-{\lambda }_1{P}^{\prime }+{\lambda }_2{P}^{\prime }}{{\lambda }_1{P}^{\prime }+{\lambda }_3{P}^{\prime }}$ by $\left(1\right)$
$=\frac{{\lambda }_2-{\lambda }_1}{{\lambda }_3-{\lambda }_1}=constant$
$\therefore \frac{t_2}{t_3}=constant$.