Question

If $x=9$ is the chord of contact of the hyperbola $x^2-y^2=9$, then the equation of the corresponding pair of tangents is

• A. $9x^2-8y^2+18x-9=0$
• B. $9x^2-8y^2-18x+9=0$
• C. $9x^2-8y^2-18x-9=0$
• D. $9x^2-8y^2+18x+9=0$

Answer 1

Answer: (B)

Let $(h,k)$ be a point whose chord of contact with respect to hyperbola $x^2-y^2=9$ is $x=9$.
We know that, chord of contact of $(h,k)$ with respect to hyperbola $x^2-y^2=9$ is $T=0$.
$\Rightarrow h\cdot x+k(-y)-9=0$
$\therefore hx-ky-9=0$
But it is the equation of the line $x=9$.
This is possible when $h=1,k=0$ (by comparing both equations).
Again equation of pair of tangents is $T^2=S{S}_1$
$\Rightarrow (x-9{)}^2=\left(x^2-y^2-9\right)\left(1^2-0^2-9\right)$
$\Rightarrow x^2-18x+81=\left(x^2-y^2-9\right)(-8)$
$\Rightarrow x^2-18x+81=-8x^2+8y^2+72$
$\Rightarrow 9x^2-8y^2-18x+9=0$