Question

Equation of chord of contact of tangents drawn from a point $(m,n)$ to hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ is given by $3x-4y=6$. Then evaluate the coordinates of $(m,n)$.

• A. $P(6,8)$
• B. $P(8,6)$
• C. $P(4,3)$
• D. $P(3,4)$

Answer 1

The correct option is B $P(8,6)$

Given: Equation of hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$; Chord of contact $3x-4y=6$

To Find: Coordinates of $P(m,n)$

Step - $1$: Recall equation of chord of contact

Step - $2$: Compare it with equation given

Step - $3$: Simplify and get values of $m,n$

We know that, the equation of the chord of contact of tangents to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ from point $P(h,k)$ is given by $\frac{hx}{a^2}-\frac{ky}{b^2}=1$.

Also, $a=4,b=3,h=m,k=n$

Chord of contact is: $\frac{mx}{16}-\frac{ny}{9}=1\cdots \left(1\right)$

Given equation of chord of contact is: $3x-4y=6$

$\Rightarrow \frac{3x}{6}-\frac{4y}{6}=1$

$\Rightarrow \frac{x}{2}-\frac{2y}{3}=1\cdots \left(2\right)$

Equation $\left(1\right)$ and $\left(2\right)$ represents the same chord of contact.

$\therefore \frac{\frac{m}{16}}{\frac{1}{2}}=\frac{\frac{-n}{9}}{-\frac{2}{3}}=1$

$\Rightarrow \frac{m}{8}=\frac{n}{6}=1$

$\Rightarrow m=8,n=6$

$P(8,6)$