if any tangent to the ellipse x2/a2+y2/b2=1 makes intercepts h and k on the axes then
h²/a²+k²/b²=1
h²/a²+k²/b²=2
a²/h²+b²/k²=1
none of the above

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Let} \mathrm{point} \mathrm{be} \mathrm{P}\left(\mathrm{acos\theta }, \mathrm{bsin\theta }\right)=\left({\mathrm{x}}_1, {\mathrm{y}}_1\right) \\ \mathrm{Equation} \mathrm{of} \mathrm{tangent} \mathrm{to} \mathrm{ellipse} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1 \mathrm{at} \left({\mathrm{x}}_1, {\mathrm{y}}_1\right) \mathrm{is} \mathrm{given} \mathrm{by}: \\ \frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}+\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}=1 \\ \Rightarrow \frac{\mathrm{x}.\mathrm{acos\theta }}{{\mathrm{a}}^2}+\frac{\mathrm{y}. \mathrm{bsin\theta }}{{\mathrm{b}}^2}=1 \\ \Rightarrow \frac{\mathrm{xcos\theta }}{\mathrm{a}}+\frac{\mathrm{ysin\theta }}{\mathrm{b}}=1 \\ \mathrm{For} \mathrm{x} \mathrm{axis} \mathrm{intercept}, \mathrm{y}=0 \\ \Rightarrow \frac{\mathrm{hcos\theta }}{\mathrm{a}}=1 \\ \Rightarrow \mathrm{cos\theta }=\frac{\mathrm{a}}{\mathrm{h}};\mathrm{equation}\left(\mathrm{i}\right) \\ \mathrm{For} \mathrm{y} \mathrm{axis} \mathrm{intercept}, \mathrm{x}=0 \\ \Rightarrow \frac{\mathrm{ksin\theta }}{\mathrm{b}}=1 \\ \Rightarrow \mathrm{sin\theta }=\frac{\mathrm{b}}{\mathrm{k}};\mathrm{equation}\left(\mathrm{ii}\right) \\ {\left(\mathrm{i}\right)}^2+{\left(\mathrm{ii}\right)}^2 \\ \Rightarrow \frac{{\mathrm{a}}^2}{{\mathrm{h}}^2}+\frac{{\mathrm{b}}^2}{{\mathrm{k}}^2}={\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta } \\ \mathbf{\Rightarrow }\frac{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{h}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{k}}^{\mathbf{2}}}\mathbf{=}\mathbf{1} \\ \mathbf{Hence}\mathbf{ }\mathbf{option}\left(\mathbf{c}\right)\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{correct}\mathbf{.} \end{gathered}$$

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