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 $PQSR$ is completed.
On the basis of the above information, answer the following questions:If $P\equiv (3,4)$, then co-ordinate 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. $(\frac{-78}{25},\frac{-51}{25})$

Answer 1

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

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