Question

$PM$ and $PN$ are the perpendiculars from any point $P$ on the rectangular hyperbola $xy=8$ to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of $(1,1)$ to director circle of the conic is

• A. $\sqrt{3}$ unit
• B. $2\sqrt{3}$ unit
• C. $2\sqrt{5}$ unit
• D. $\sqrt{2}$ unit

Answer 1

The correct option is D $\sqrt{2}$ unit



Here, $OMPN$ is rectangle.
$P\equiv (2\sqrt{2}t,\frac{2\sqrt{2}}{t})$

Mid point of $MN=OP$
$\therefore (h,k)=(\sqrt{2}t,\frac{\sqrt{2}}{t})$
Thus $h\times k=\sqrt{2}t\cdot \frac{\sqrt{2}}{t}=2$
Thus locus is the rectangular hyperbola $xy=2$
The director circle to this rectangular hyperbola is the point circle with center $(0,0)$.
Thus distance between $(0,0)$ and $(1,1)$ is $\sqrt{2}$ unit