Tangents are drawn from any point on the hyperbola $\frac{{\mathrm{x}}^2}{9}-\frac{{\mathrm{y}}^2}{4}=1$ to the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=9$. If the locus of the mid-point of the chord of contact is $\mathrm{a}{({\mathrm{x}}^2+{\mathrm{y}}^2)}^2 = {\mathrm{bx}}^2 – {\mathrm{cy}}^2$, then the value of ${\mathrm{a}}^2 +{\mathrm{b}}^2+{\mathrm{c}}^2 =$

$$\begin{gathered} \mathrm{let} \mathrm{P}=(3\mathrm{sec\theta }, 2\mathrm{tan\theta }) \mathrm{be} \mathrm{any} \mathrm{point} \mathrm{on} \mathrm{given} \mathrm{hyperbola} \\ \mathrm{Now} equation \mathrm{of} \tan gent of \mathrm{P} \mathrm{w}.\mathrm{r}. \mathrm{to} {\mathrm{x}}^2+{\mathrm{y}}^2=9 \mathrm{is} 3\mathrm{x} \mathrm{sec\theta }+2\mathrm{y} \mathrm{tan\theta }=9 — ① \\ \mathrm{let} ({\mathrm{x}}_1, {\mathrm{y}}_1) \mathrm{be} \mathrm{mid} \mathrm{point} \mathrm{of} \mathrm{chord} \mathrm{of} \mathrm{contact} \mathrm{then} \mathrm{equation} \mathrm{of} \mathrm{chord} \mathrm{is} {\mathrm{xx}}_1+{\mathrm{yy}}_1={{\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2 - ② \\ \mathrm{Since} ① \& ② \mathrm{represents} \mathrm{same} \mathrm{line} \mathrm{then}. \\ \Rightarrow \frac{3\mathrm{sec\theta }}{{\mathrm{x}}_1}=\frac{2\mathrm{tan\theta }}{{\mathrm{y}}_1}=\frac{9}{{{{\mathrm{x}}_1}^2}_{+}{{{{\mathrm{y}}_1}^2}^{ }}_{ }} \\ \Rightarrow sec\theta =\frac{9x_1}{3\left({{{\mathrm{x}}_1}^2}_{+}{{\mathrm{y}}_1}^2\right)} and \tan \theta =\frac{9y_1}{2\left({{{\mathrm{x}}_1}^2}_{+}{{\mathrm{y}}_1}^2\right)} \\ \Rightarrow {\sec }^2\mathrm{\theta }-{\tan }^2\mathrm{\theta }=1 \mathrm{then} \\ \frac{81{{\mathrm{x}}_1}^2}{9{\left({{{\mathrm{x}}_1}^2+}_{ }{{\mathrm{y}}_1}^2\right)}^2}-\frac{81{y_1}^2}{4{\left({{{\mathrm{x}}_1}^2+}_{ }{{\mathrm{y}}_1}^2\right)}^2}=1 \\ \frac{{{\mathrm{x}}_1}^2}{9}-\frac{{{\mathrm{y}}_1}^2}{4}={\frac{({{\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2)}{81}}^2 \\ {\mathrm{locus} \mathrm{is} 36\mathrm{x}}^2-81{\mathrm{y}}^2=4{({\mathrm{x}}^2+{\mathrm{y}}^2)}^2 \mathrm{By} \mathrm{comparision} \mathrm{a}=4, \mathrm{b}=36, \mathrm{c}=81 \\ \mathrm{Now} {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=7873 \end{gathered}$$