The locus of mid-points of points of intersection of $x \cos \theta+y \sin \theta=1$ with the coordinate…
The locus of mid-points of points of intersection of $x \cos \theta+y \sin \theta=1$ with the coordinate axes is
$x^2+y^2=4$
$\frac{1}{x^2}+\frac{1}{y^2}= {4}$
$\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{2}$
$x^2+y^2=2$
Solution
Let the line cut the axes in $A$ and $B$ and if $(h, k)$ be the mid-point of $A B$, then
$2 h=\frac{1}{\cos \theta}, 2 k=\frac{1}{\sin \theta}$
In order to find the locus, eleminate the variable $\theta$ by $\cos ^2 \theta+\sin ^2 \theta=1$
$\frac{1}{4 h^2}+\frac{1}{4 k^2}=1 \Rightarrow \frac{1}{x^2}+\frac{1}{y^2}=4$