Question

Find the locus of the midpoint of the portion of the line $x\cos \alpha +y\sin \alpha =p$ which is intercepted between the axes.

Answer 1

Let the line cut the axes in $A$ and $B$ and if $(h,k)$ be the midpoints of $AB$, then
$2h=\frac{p}{\cos \theta },2k=\frac{p}{\sin \theta }$
In order to find the locus, eliminate the variable $\theta$ by ${\cos }^2\theta +{\sin }^2\theta =1$
$\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$