Question

If line $y=2x+10$ cuts the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ at $P$ and $Q$ , then point of intersection of tangents at $P$ and $Q$ to hyperbola is

• A. $\left[-\frac{9}{5},-\frac{2}{5}\right]$
• B. $\left[\frac{9}{5},\frac{2}{5}\right]$
• C. $\left[\frac{5}{9},\frac{5}{2}\right]$
• D. $\left[-\frac{5}{9},-\frac{2}{9}\right]$

Answer 1

Answer: (A)

$\left[-\frac{9}{5},-\frac{2}{5}\right]$

The equation $y=2x+10$ becomes the chord of contact to the Required point

Let the required point be $(x_1,y_1)$

The chord of contact is given as $\frac{x{x}_1}{9}-\frac{y{y}_1}{4}=1y=\frac{4}{y_1}(\frac{x{x}_1}{9}-1)\cdots \left(1\right)$

Comparing it with $y=2x+10$

$-\frac{4}{y_1}=10y_1=\frac{-2}{5}\frac{4x_1}{9y_1}=2x_1=\frac{-9}{5}$