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

$\text{ Let a pair of tangents be drawn from the point }\left(x_1,y_1\right)\text{ to the hyperbola}$

$x^2-y^2=9$

Then the chord of contact will be 

${\mathrm{xx}}_1-{\mathrm{yy}}_1=9-----\left(\mathrm{i}\right)$

But the given chord of contact is 

$x=9-----\left(ii\right)$

As (i) and (ii) represent the same line, these equations should be identical and, hence

$\frac{x_1}{1}=-\frac{y_1}{0}=\frac{9}{9}\text{ or }{x}_1=1,y_1=0$

Therefore, the equation of pair of tangents drawn from (1,0) to $x^2-y^2=9\text{ is }$

$\begin{array}{l}\left({\mathrm{x}}^2-{\mathrm{y}}^2-9\right)\left(1^2-0^2-9\right)=(\mathrm{x}\cdot 1-\mathrm{y}\cdot 0-9{)}^2\left(\text{ Using }{\mathrm{SS}}_1={\mathrm{T}}^2\right)\\ \text{ or }\left({\mathrm{x}}^2-{\mathrm{y}}^2-9\right)(-8)=(\mathrm{x}-9{)}^2\\ \text{ or }-8{\mathrm{x}}^2+8{\mathrm{y}}^2+72={\mathrm{x}}^2-18\mathrm{x}+81\\ \text{ or }9{\mathrm{x}}^2-8{\mathrm{y}}^2-18\mathrm{x}+9=0\end{array}$