First slide

Hyperbola in conic sections

Question

$H_{1} : x y = c^{2} and H_{2} : x y = k^{2}$ are two different hyperbolas. From a point on $H_{1}$ , tangents are drawn to $H_{2}$ .Area of the triangle formed by the chord of contact and the asymptote to $H_{2}$ is

Moderate

A

$\frac{k^{2}}{c^{2}}$
B

$\frac{k^{4}}{c^{2}}$
C

$\frac{2 k^{4}}{c^{2}}$
D

none of these

Solution

Equation of the chord of contact from $(c t, \frac{c}{t}) on H_{1}$ $to H_{2} is (c t) x + (\frac{c}{t}) y = 2 k^{2} .$
Asymptote of $H_{2}$ are $x = 0$ and $y = 0$ and the chord meets
these asymptotes at A $(\frac{2 k^{2}}{c t}, 0) and B (0, \frac{2 k^{2} t}{c})$
Area of the required triangle is $\frac{1}{2} |\frac{2 k^{2}}{c t}| |\frac{2 k^{2} t}{c}| = \frac{2 k^{4}}{c^{2}}$

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