A variable tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ makes intercepts on both the axes. The locus of the middle point of the portion of the tangent between the coordinate axes is

Given:
The equation of ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Equation of tangent is $\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$.

The tangent meets the axes at $A\left(\frac{a}{\cos \theta },0\right)$ and $B\left(0,\frac{b}{\sin \theta }\right)$.

Let $\left(x_1,y_1\right)$ be the midpoint of $AB$.

$\begin{array}{l}\left(\frac{{\displaystyle \frac{a}{\cos \theta }}+0}{2}, \frac{{\displaystyle 0+\frac{b}{\sin \theta }}}{2}\right)=\left(x_1,y_1\right)\end{array}$

$\begin{array}{l}\Rightarrow \frac{a}{\cos \theta }=2x_1\Rightarrow \cos \theta =\frac{a}{2x_1}\end{array}$

$\begin{array}{l}\Rightarrow \frac{b}{\sin \theta }=2y_1\Rightarrow \sin \theta =\frac{b}{2y_1}\end{array}$

Now, ${\sin }^2\theta +{\cos }^2\theta =1$

$\Rightarrow {\left(\frac{b}{2y_1}\right)}^2+{\left(\frac{a}{2x_1}\right)}^2=1$

$\Rightarrow \frac{b^2}{4y_1^2}+\frac{a^2}{4x_1^2}=1$

$\Rightarrow \frac{a^2}{x_1^2}+\frac{b^2}{y_1^2}=4$
$\therefore$ Locus of mid-point is $\frac{a^2}{x^2}+\frac{b^2}{y^2}=4$.