Question

From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$ . Then, area cut-off by the chord of contact on the asymptotes is equal to

• A. a/2 sq unit
• B. ab sq unit
• C. 2ab sq unit
• D. 4ab sq unit

Answer 1

The correct option is D 4ab sq unit

Let P$(x_1,y_1)$ be a point on the hyperbola $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
The chord of contact of tangents from P to the hyperbola is given by $\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=1$ …… (i)
The equation of the asymptotes are $\frac{x}{a}-\frac{y}{b}=0$
and $\frac{x}{a}+\frac{y}{b}=0$
The points of intersection of Equation (i) with the two asymptotes are given by
$x_1=\frac{2a}{\frac{x_1}{a}+\frac{y_1}{b}},y_1=\frac{2a}{\frac{x_1}{a}+\frac{y_1}{b}}$
$x_2=\frac{2a}{\frac{x_1}{a}+\frac{y_1}{b}},y_2=\frac{2a}{\frac{x_1}{a}+\frac{y_1}{b}}$
Area of the triangle = $\frac{1}{2}(x_1{x}_2-x_2{y}_1)$
$=\frac{1}{2}\left|(-\frac{4ab\times 2}{\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}})\right|$