Question

Find the locus of the middle point of the portion of the line $x\cos \alpha +y\sin \alpha =p$ which is intercepted between the axes, given that $p$ remains constant.

Answer 1

$x\cos \alpha +y\sin \alpha =p$

$x,y$ intercepts are $\frac{p}{\cos \alpha },\frac{p}{\sin \alpha }$

Let $(h,k)$ be the mid point of the line intercepted between the two axes.

$\therefore (h,k)=(\frac{p}{2\cos \alpha },\frac{p}{2\sin \alpha })h=\frac{p}{2\cos \alpha },k=\frac{p}{2\sin \alpha }\Rightarrow \frac{1}{h^2}+\frac{1}{k^2}=\frac{4}{p^2}$

The locus is $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$