Mathematics - Ellipse Question with Solution | TestHub

The equation of the tangent at $(5\cos \theta ,2\sin \theta )$ is $\frac{x}{5}\cos \theta +\frac{y}{2}\sin \theta =1$ If it is a tangent to the circle then $\frac{1}{\sqrt{\frac{{\cos }^2\theta }{25}+\frac{{\sin }^2\theta }{4}}}=4$

$$\Rightarrow \cos \theta =\frac{10}{4\sqrt{7}},\sin \theta =\frac{\sqrt{3}}{2\sqrt{7}}$$

Let A and B be the points where the tangent meets the coordinate axis then

$$\begin{array}{rlr}& A\left(\frac{5}{\cos \theta },0\right),B\left(0,\frac{2}{\sin \theta }\right)& L=\sqrt{\frac{25}{{\cos }^2\theta }+\frac{4}{{\sin }^2\theta }}=\frac{14}{\sqrt{3}}\end{array}$$