Question

The perpendicular PL, PM are drawn from any point P on the rectangular hyperbola xy=25 to the asymptotes the locus of the mid point of OP is curve with eccentricity

• A. An ellipse with e = $\sqrt{2}$
• B. An hyperbola with e =$\sqrt{2}$
• C. A parabola e = 1/$\sqrt{2}$
• D. None of these

Answer 1

Answer: (B)

An hyperbola with e =$\sqrt{2}$

Answer 2

Let R be the mid point of OP

As xy = 5² = c²

∴ P(x, y) = $\left( 5t,\frac{5}{t} \right)$ and OMPL is a rectangle

Coordinate of R = $\left( \frac{5t}{2},\mspace{6mu}\frac{5}{2t} \right)$

Now locus of Q is xy = $\frac{5t}{2}x\frac{5}{2t} = \frac{25}{4}$

Which is a rectangular hyperbola, whose eccentricity is given e² = $\frac{a^{2} + b^{2}}{a^{2}}\mspace{6mu} = \mspace{6mu}\frac{a^{2} + b^{2}}{b^{2}}\mspace{6mu}(\because = a = b)$

∴ e² = $\frac{2a^{2}}{a^{2}} = 2$

⇒ e² = 2 ∴ e = $\sqrt{2}$