Question

The point of intersection of 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 $lx + my + n = 0$ is

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

Answer 1

Answer: (A)

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

Answer 2

Let $(x_{1},y_{1})$ be the required point. Then the equation of the chord of contact of tangents drawn from $(x_{1},y_{1})$ to the given hyperbola is $\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1$ ......(i)

The given line is $lx + my + n = 0$ .....(ii)

Equation (i) and (ii) represent the same line

$\therefore$ $\frac{x_{1}}{a^{2}l} = - \frac{y_{1}}{b^{2}m} = \frac{1}{- h}$ ⇒ $x_{1} = \frac{- a^{2}l}{n},y_{1} = \frac{b^{2}m}{n}$; Hence the required point is $\left( - \frac{a^{2}l}{n},\frac{b^{2}m}{n} \right)$