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Q.
If the tangent line to an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ cuts intercepts $h$ and $k$ from axes, then $\frac{a^{2}}{h^{2}}+\frac{b^{2}}{k^{2}}=$
Conic Sections
Solution:
Let the equation of tangent at the point $\left(x_{1}, y_{1}\right)$ be
$\frac{x x_{1}}{a^{2}}+\frac{y y_{1}}{b^{2}}=1$
$ \Rightarrow \frac{x}{\frac{a^{2}}{x_{1}}}+\frac{y}{\frac{b^{2}}{y_{1}}}=1$
which meets axes at $\left(\frac{a^{2}}{x_{1}}, 0\right)$ and $\left(0, \frac{b^{2}}{y_{1}}\right)$
but $\frac{a^{2}}{x_{1}}=h$ and $\frac{b^{2}}{y_{1}}=k$ (given)
$\Rightarrow x_{1}=\frac{a^{2}}{h}, y_{1}=\frac{b^{2}}{k}$
$ \Rightarrow \frac{x_{1}^{2}}{a^{2}}+\frac{y_{1}^{2}}{b^{2}}=1 $
$\therefore \frac{a^{2}}{h^{2}}+\frac{b^{2}}{k^{2}}=1$