Question

From any point on a hyperbola xy = c² tangents are drawn to another hyperbola xy = a² which has the same asymptotes. Then the chord of contact cuts off a constant area from the asymptotes:

• A. $\frac{2a^{2}}{c^{2}}$
• B. $\frac{a^{2}}{c^{2}}$
• C. $\frac{2c^{2}}{a^{2}}$
• D. $\frac{c^{2}}{a^{2}}$

Answer 1

Answer: (A)

$\frac{2a^{2}}{c^{2}}$

Answer 2

Let xy = c² be the rectangular hyperbola referred to its asymptotes as the coordinate axis.

Let P(h, k) be a point on

xy = c² ... (i)

tangents are drawn from P(h, k) to the rectangular hyperbola xy = a²

\ equation of chord of contact

Ž kx + hy = 2a²

This cuts the coordinate axis at A $\left( \frac{2a^{2}}{k},0 \right)$

and B $\left( 0,\frac{2a^{2}}{h} \right)$

\ Area of D OAB = $\frac{1}{2}$ $\left( \frac{2a^{2}}{k} \times \frac{2a^{2}}{h} \right)$

=$\frac{2a^{4}}{kh}$= $\frac{2a^{2}}{c^{2}}$