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 $T=0$.
$\Rightarrow h \cdot x+k(-y)-9=0$
$\therefore h x-k y-9=0$
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}-18 x+81=\left(x^{2}-y^{2}-9\right)(-8) $
$\Rightarrow x^{2}-18 x+81=-8 x^{2}+8 y^{2}+72$
$\Rightarrow 9 x^{2}-8 y^{2}-18 x+9=0$