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 $L\equiv y=4,$ then the locus of the circumcentre of $△PQR$ is

• A. $y-2=0$
• B. $x-2=0$
• C. $y+2=0$
• D. $x+2=0$

Answer 1

The correct option is A $y-2=0$

∵ Length of tangents from a point to circle are equal.

PQ=PR

Then parallelogram PQRS is rhombus.

∴ Mid-point of QR= midpoint of PS

and QR⊥PS

∴S is the mirror image of P w.r.t. QR

∵L≡y-4=0

Let P=(0,4)

∵∠PQO=∠PRO= $\frac{\pi }{2}$

​

∴OP is diameter of circumcircle PQR then centre is

(0,4)

$x=0$

and y=4, then h=2

then y=2