Question

The coordinates of the point of intersection of the tangents drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at the points where it is intersected by the line $lx+my+n=0$ are

• A. $(\frac{a^{2}l}{n},-\frac{b^{2}m}{n})$
• B. $(-\frac{a^{2}l}{n},\frac{b^{2}m}{n})$
• C. $(\frac{a^{2}l}{n},\frac{b^{2}m}{n})$
• D. $(-\frac{a^{2}l}{n},-\frac{b^{2}m}{n})$

Answer 1

Answer: (B)

$(-\frac{a^{2}l}{n},\frac{b^{2}m}{n})$

Let $(x_1,y_1)$ be the required points.
Then, the equation of the chord of contact of the tangents draw from $(x_1,y_1)$ to the given hyperbola is
$\frac{{xx}_1}{a^2}-\frac{{yy}_1}{b^2}=1\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(i\right)$
The given line is $lx+my+n=0\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(ii\right)$
Equations $\left(i\right)$ and $\left(ii\right)$ represent the same line
$\therefore \frac{x_1}{a^{2}l}=\frac{{-y}_1}{b^{2}m}=\frac{1}{-n}$
$\Rightarrow x_1=\frac{-a^{2}l}{n},y_1=\frac{b^{2}m}{n}$
Therefore,, the required points is $(\frac{-a^{2}l}{n},\frac{b^{2}m}{n})$
Hence, option 'B' is correct.