Question

lf $P$ is a variable point and $PQ$ and $PR$ are the tangents drawn to the circle $x^2+y^2=9$. Let $QR$ be always touching the circle $x^2+y^2=4$, then locus of $P$ is

• A. $4x^2+4y^2=81$
• B. $2x^2+2y^2=81$
• C. $x^2+y^2=16$
• D. $x^2+y^2=4$

Answer 1

Answer: (A)

$4x^2+4y^2=81$

The diagram has 2 circles. The outer one is $x^2+y^2=9$ and the inner one is $x^2+y^2=4$.

Let $P(h,k)$ be the required point from where tangents $PQ$ and $QR$ are drawn and $A$ be the point of tangency of $QR$ with the inner circle.

Hence, equation of $QR$ can be found as $xh+yk-9=0$. (from pole and polar concept)

Now, if this line has the shortest distance of 2 units from origin, it is a tangent to $x^2+y^2=4$.

$\therefore 2=\left|\frac{0\left(h\right)+0\left(k\right)-9}{\sqrt{h^2+k^2}}\right|$

$\therefore 2=\left|\frac{-9}{\sqrt{h^2+k^2}}\right|$

$\therefore 4=\frac{81}{h^2+k^2}$ (squaring both sides)

$\therefore 4(h^2+k^2)=81$

Hence, the required locus is $4x^2+4y^2=81$