Question

If $C_1,C_2$ and $C_3$ belong to a family of circles through the points $(x_1,y_1)$ and $(x_2,y_2)$, prove that the ratio of the lengths of the tangents from any point on $C_1$ to the circle $C_2$ and $C_3$ is constant.

Answer 1

The equation of orders through

$(x_1,y_1)$ and $(x_2,y_2)$

$\Rightarrow (x-x_1)(x-x_2)+(y-y_1)(y-y_2)+{\lambda }_r\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|=0$

$(r=1,2,3)$

Let $(h,k)$ be a point on circle $c_1$

Then,$\phi(h,k)+\lambda p(h,k)=0 \rightatrrow(1)$

Where $\varphi (h,k)=(h-x_1)(h-x_2)+(k-y_1)(k-y_2)$

and $p(h,k)=\left|\begin{array}{ccc}h& k& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|$

Let $T_2$ be the length of tangent from $(h,k)$ to C_{2}$and$T_{3}$bethelengthoftangentfrom$(h,k)$to$C_{3}$,

then :

$T_2=\sqrt{\varphi (h,k)+{\lambda }_{2}p(h,k)}$

$T_3=\sqrt{\varphi (h,k)+{\lambda }_{3}p(h,k)}$

($\because$ The length of tangent from $(x_1,y_1)$ to the circle $s=0$ is$\sqrt{s_{11}}$)

$\Rightarrow \frac{T_2}{T_3}=\frac{\sqrt{\varphi (h,k)+{\lambda }_2(h,k)}}{\sqrt{\varphi (h,k)+{\lambda }_{3}p(h,k)}}$

$=\sqrt{\frac{({\lambda }_2-{\lambda }_1)p(h,k)}{({\lambda }_3-{\lambda }_1)p(h,k)}}$

($\because \varphi (h,k)=-{\lambda }_{1}p\left(hk\right)$ (from $\left(1\right)$))

$=\sqrt{\frac{({\lambda }_2-{\lambda }_{11})}{({\lambda }_3-{\lambda }_1)}}$

$\Rightarrow \frac{T_2}{T_3}=constant$