Question

If $PN$ is the perpendicular from a point $P$ on a rectangular hyperbola to its asymptotes, the locus of the mid-point of $PN$ is

• A. circle
• B. parabola
• C. ellipse
• D. hyperbola

Answer 1

The correct option is D hyperbola

Let $xy=c^2$ be the rectangular hyperbola and let $P(x_1,y_1)$ be apoint on it. Let $Q(h.k)$ be the
midpoint of $PN$. Then the coordinates of $Q$ are $\left(x_1,y_1/2\right)$
$\therefore x_1=h\frac{y_1}{2}=k\Rightarrow x_1=h,y_1=2k$
But $(x_1,y_1)$ lies on $xy=c^2$
$\therefore h\left(2k\right)=c^2\Rightarrow hk=\frac{c^2}{2}$
Hence the locus of $(h,k)$ is $xy=\frac{c^2}{2}$which is a hyperbola.