Question

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

• A. Circle
• B. Parabola
• C. Ellipse
• D. Hyperbola

Answer 1

Answer: (D)

Hyperbola

Answer 2

Let $xy = c^{2}$ be the rectangular hyperbola, and let $P(x_{1},y_{1})$ be a point on it. Let $Q(h,k)$ be the mid-point of PN. Then the coordinates of Q are $\left( x_{1},\frac{y_{1}}{2} \right)$.

$\therefore$ $x_{1} = h$ and $\frac{y_{1}}{2} = k$ ⇒ $x_{1} = h$ and $y_{1} = 2k$

But $(x_{1},y_{1})$ lies on xy = c².

$\therefore h.(2k) = c^{2}$ ⇒ $hk \Rightarrow c^{2}/2$

Hence, the locus of $(h,k)$ is $xy = c^{2}/2$, which is a hyperbola