Question

The chords passing through $\left(2,1\right)$ intersect the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ at $A$ and $B.$ The locus of the point of intersection of tangents at $A$ and $B$ on the hyperbola is

• A. $x-y=1$
• B. $x+y=3$
• C. $9x-8y=72$
• D. $9x+8y=7$

Answer 1

Answer: (C)

Let, the point of intersection of the tangents at $A$ & $B$ is $\left(h , k\right).$
Then, the equation of $AB$ is $\frac{x h}{16}-\frac{y k}{9}=1$ { $\because T=0$ is the chord of contact}
Now, $\frac{x h}{16}-\frac{y k}{9}=1$ passes through $\left(2,1\right)$
Hence, $\frac{2 h}{16}-\frac{k}{9}=1$
$\Rightarrow 9h-8k=72$
$\Rightarrow $ the locus is $9x-8y=72$