If $\mathrm{x} = 9$ is a chord of contact of the hyperbola ${\mathrm{x}}^2–{\mathrm{y}}^2=9,$ then the equation of the tangent at one of the points of contact is

$$\begin{gathered} \mathrm{Given} \mathrm{hyperbola} {\mathrm{x}}^2-{\mathrm{y}}^2=9 — ① \\ and given chord of contact \mathrm{x}=9 — ② \\ \mathrm{By} \mathrm{solving} ① \& ② \mathrm{we} \mathrm{get} \mathrm{point} \mathrm{of} \mathrm{contact} \\ sub x=9 in \left(1\right) \\ 81-y^2=9 \\ \Rightarrow y^2=72 \\ \Rightarrow y=\pm 6\sqrt{2} \\ \therefore \mathrm{P}=(9, \pm 6\sqrt{2}) \\ \mathrm{Now} \mathrm{equation} \mathrm{of} \mathrm{tangent} \mathrm{at} \mathrm{P}=(9, 6\sqrt{2}) \mathrm{is} {\mathrm{S}}_1=0 \\ \Rightarrow \mathrm{x}\left(9\right)-\mathrm{y}\left(6\sqrt{2}\right)-9=0 \\ \Rightarrow 3\mathrm{x}-2\sqrt{2}\mathrm{y}-3=0 \end{gathered}$$