Question

Locus of a point, whose chord of contact with respect to the circle $x^2+y^2=4$ is a tangent to hyperbola $xy=1$ is :

• A. $xy=4$
• B. $x^2-y^2=6$
• C. $xy=8$
• D. $x^2-y^2=16$

Answer 1

The correct option is A $xy=4$

Let point be $P(h,k)$
chord of contact of from $P$ to $x^2+y^2=4$ is
$hx+ky=4\cdots \left(1\right)$
$\left(1\right)$ is tangent to $xy=1$
So solving equation $\left(1\right)$ with $xy=1$, we have
$x\cdot \left(\frac{4-hx}{k}\right)=1$
$\Rightarrow 4x-h{x}^2=k$
$\Rightarrow h{x}^2-4x+k=0$
Discrimant of the above equation must be zero for the equation to touch at one point.
$\therefore 16-4hk=0$
$\Rightarrow xy=4$
Hence, locus is a rectangular hyperbola.