Question

$P$ is a variable point on the line $L=0$. Tangents are drawn to the circle $x^2+y^2=4$ from $P$ to touch it at $Q$ and $R$. The parallelogram $PQRS$ is completed.

If $P\equiv (3,4)$, then coordinate of $S$ is

• A. $(-\frac{46}{25},-\frac{63}{25})$
• B. $(-\frac{51}{25},-\frac{68}{25})$
• C. $(-\frac{46}{25},-\frac{68}{25})$
• D. $(-\frac{68}{25},-\frac{51}{25})$

Answer 1

The correct option is B $(-\frac{51}{25},-\frac{68}{25})$


$\because P\equiv (3,4)$
$\therefore$ Equation of $QR$ is
$T=0\Rightarrow 3x+4y=4\dots \left(1\right)$
Let $S\equiv (x_1,y_1)$
$\because S$ is the mirror image of $P$ w.r.t equation$\left(1\right)$, then
$\frac{x_1-3}{3}=\frac{y_1-4}{4}=\frac{-2(3\times 3+4\times 4-4)}{3^2+4^2}=-\frac{42}{25}$
$\therefore x_1=-\frac{51}{25},y_1=-\frac{68}{25}\Rightarrow S\equiv (-\frac{51}{25},-\frac{68}{25})$