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 $L\equiv 2x+y-6=0$, then the locus of circumcentre of $△PQR$ is

• A. $2x-y=4$
• B. $2x+y=3$
• C. $x-2y=4$
• D. $x+2y=3$

Answer 1

The correct option is B $2x+y=3$

$\because PQ=PR$ i.e. parallelogram $PQRS$ is a rhombus
$\therefore$ Mid point of $OR=$Mid point of $PS$ and $QR\perp PS$

$\therefore S$ is the mirror image of $P$ w.r.t. $QR$
$\because L\equiv 2x+y=6$
Let $P\equiv (k,6-2k)$
$\because \angle PQO=\angle PRO=\frac{\pi }{2}$
$\therefore OP$ is diameter of circumcircle $PQR$, then centre is mid point of $OP$
$=(\frac{k}{2},3-k)$
$\therefore x=\frac{k}{2}\Rightarrow k=2x\Rightarrow y=3-k\Rightarrow 2x+y=3$