Equation of $$AB$$ is

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

$$ \Rightarrow {{x\cos \alpha } \over p} + {{y\sin \alpha } \over p} = 1;$$

$$ \Rightarrow {x \over {p/\cos \alpha }} + {y \over {p/\sin \alpha }} = 1$$

So co-ordinates of $$A$$ and $$B$$ are

$$\left( {{p \over {\cos \alpha }},0} \right)$$ and $$\left( {0,{p \over {\sin \alpha }}} \right);$$

So coordinates of midpoint of $$AB$$ are

$$\left( {{p \over {2\cos \,\alpha }},{p \over {2\sin \alpha }}} \right) = \left( {{x_1},{y_1}} \right)\left( {let} \right);$$

$${x_1} = {p \over {2\,\cos \,\alpha }}\,\,\& \,\,{y_1} = {p \over {2\sin \alpha }};$$

$$ \Rightarrow \cos \alpha = p/2{x_1}$$ and $$\sin \alpha = p/2{y_1};$$

$${\cos ^2}\alpha + {\sin ^2}\alpha = 1 \Rightarrow {{{p^2}} \over 4}\left( {{1 \over {{x_1}^2}} + {1 \over {{y_1}^2}}} \right) = 1$$

Locus of $$\left( {{x_1},{y_1}} \right)$$ is $${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}.$$