the slope of a common tangent to the ellipse x²/a² + y² /b² =1 and a concentric circle of radius r is ( r² -b² )^1/2/(a²-r²)

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1 \\ \mathrm{Line} \mathrm{y}=\mathrm{mx}+\mathrm{c} \mathrm{is} \mathrm{tangent} \mathrm{to} \mathrm{this} \mathrm{ellipse} \mathrm{if} \\ {\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2....\left(\mathrm{i}\right) \\ \mathrm{Centre} \mathrm{of} \mathrm{ellipse} \mathrm{is} \left(0,0\right) \mathrm{and} \mathrm{as} \mathrm{circle} \mathrm{is} \mathrm{concentric} \mathrm{hence} \mathrm{its} \mathrm{centre} \mathrm{will} \mathrm{also} \\ \mathrm{be} \left(0,0\right). \\ \mathrm{Radius} \mathrm{is} \mathrm{r}, \mathrm{hence} \\ {\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{r}}^2 \\ \mathrm{Line} \mathrm{y}=\mathrm{mx}+\mathrm{c} \mathrm{is} \mathrm{tangent} \mathrm{to} \mathrm{this} \mathrm{circle} \mathrm{if} \\ {\mathrm{c}}^2={\mathrm{r}}^2\left(1+{\mathrm{m}}^2\right)....\left(\mathrm{ii}\right) \\ \mathrm{By} \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right) \\ {\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2={\mathrm{r}}^2\left(1+{\mathrm{m}}^2\right) \\ \Rightarrow {\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2={\mathrm{r}}^2+{\mathrm{r}}^2{\mathrm{m}}^2 \\ \Rightarrow {\mathrm{a}}^2{\mathrm{m}}^2-{\mathrm{r}}^2{\mathrm{m}}^2={\mathrm{r}}^2-{\mathrm{b}}^2 \\ \Rightarrow {\mathrm{m}}^2\left({\mathrm{a}}^2-{\mathrm{r}}^2\right)={\mathrm{r}}^2-{\mathrm{b}}^2 \\ \Rightarrow {\mathrm{m}}^2=\frac{{\mathrm{r}}^2-{\mathrm{b}}^2}{{\mathrm{a}}^2-{\mathrm{r}}^2} \\ \mathbf{\Rightarrow }\mathbf{m}\mathbf{=}\sqrt{\frac{{\mathbf{r}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{r}}^{\mathbf{2}}}} \end{gathered}$$

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