The locus of the midpoints of the chords of the circle${\mathrm{x}}^2 + {\mathrm{y}}^2 = 16$ which are tangents to the hyperbola $9{\mathrm{x}}^2 – 16{\mathrm{y}}^2 = 144$ is ${({\mathrm{x}}^2 + {\mathrm{y}}^2)}^2 = {\mathrm{ax}}^2 + {\mathrm{by}}^2$ then a + b =

$$\begin{gathered} \mathrm{Given} \mathrm{hyperbola} \frac{{\mathrm{x}}^2}{16}-\frac{{\mathrm{y}}^2}{9}=1 \\ \mathrm{where} {\mathrm{a}}^2=16, {\mathrm{b}}^2=9 \\ \mathrm{let} (\mathrm{h}, \mathrm{k}) \mathrm{be} \mathrm{mid} \mathrm{point} \mathrm{of} c\mathrm{hord} \mathrm{of} \mathrm{circle} {\mathrm{x}}^2+{\mathrm{y}}^2=16 \\ \mathrm{Now} \mathrm{equation} \mathrm{of} \mathrm{chord} \mathrm{is} {\mathrm{S}}_1={\mathrm{S}}_{11} \\ \Rightarrow \mathrm{x}\left(\mathrm{h}\right)+\mathrm{y}\left(\mathrm{k}\right)-16={\mathrm{h}}^2+{\mathrm{k}}^2-16 \\ \Rightarrow \mathrm{hx}+\mathrm{ky}={\mathrm{h}}^2+{\mathrm{k}}^2 \\ \Rightarrow \mathrm{y}=\left(\frac{-\mathrm{h}}{\mathrm{k}}\right)\mathrm{x}+\frac{{\mathrm{h}}^2+{\mathrm{k}}^2}{\mathrm{k}} \\ \mathrm{If} \mathrm{it} \mathrm{is} \mathrm{a} \mathrm{tangent} \mathrm{to} \mathrm{the} \mathrm{given} \mathrm{hyperbola} \mathrm{then} \\ {c}^2={\mathrm{a}}^2{\mathrm{m}}^2-{\mathrm{b}}^2 \\ \Rightarrow \frac{{({\mathrm{h}}^2+{\mathrm{k}}^2)}^2}{{\mathrm{k}}^2}=16\left(\frac{{\mathrm{h}}^2}{{\mathrm{k}}^2}\right)-9 \\ \Rightarrow \frac{{({\mathrm{h}}^2+{\mathrm{k}}^2)}^2}{{\mathrm{k}}^2}=\frac{16{\mathrm{h}}^2-9{\mathrm{k}}^2}{{\mathrm{k}}^2} \\ \Rightarrow 16{\mathrm{h}}^2-9{\mathrm{k}}^2={({\mathrm{h}}^2+{\mathrm{k}}^2)}^2 \\ \therefore \mathrm{Locus} \mathrm{is} 16{\mathrm{x}}^2-9{\mathrm{y}}^2={({\mathrm{x}}^2+{\mathrm{y}}^2)}^2 \\ \mathrm{By} \mathrm{emparision} \mathrm{a}=16, \mathrm{b}=-9 \\ \mathrm{Now} \mathrm{a}+\mathrm{b}=16-9=7 \end{gathered}$$