Thank you for reporting, we will resolve it shortly
Q.
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
Conic Sections
Solution:
Let a pair of tangents be drawn from the point
$\left(x_{1}, y_{1}\right)$ to the hyperbola
$x^2 - y^2 = 9$
Then the chord of contact will be
$x x_{1}-y y_{1}=9$ ... (i)
But the given chord of contact is
$x=9$ .... (ii)
As (i) and (ii) represent the same line, these equations should be identical and, hence,
$\frac{x_{1}}{1}=-\frac{y_{1}}{0}=\frac{9}{9} $
or $ x_{1}=1, y_{1}=0$
Therefore, the equation of pair of tangents drawn from $(1,0)$ to $x^{2}-y^{2}=9$ is
$\left.\left(x^{2}-y^{2}-9\right)\left(1^{2}-0^{2}-9\right)=(x \cdot 1-y \cdot 0-9)^{2} \text { (Using } S S_{1}=T^{2}\right)$
or $\left(x^{2}-y^{2}-9\right)(-8)=(x-9)^{2}$
or $-8 x^{2}+8 y^{2}+72=x^{2}-18 x+81 $
or $ 9 x^{2}-8 y^{2}-18 x+9=0$