Question

Find the locus of the middle points of the chords of the circle $x^2+y^2=a^2$ which subtend a right angle at the centre.

Answer 1

Equation of the chord $AB$ of the circle
$x^2+y^2=a^2$
Whose mid-point is $(h,k)$ is $hx+ky=h^2+k^2$.
$\angle ACB=0^{\circ }\left(given\right),D(h,k)$ is the mid-point of $AB$
$\therefore \angle ACD=\angle BCD={45}^{\circ }$
$CD=a\cos {45}^{\circ }=\frac{a}{\sqrt{2}}$ or $C{D}^2=\frac{a^2}{2}$
or $h^2+k^2=\frac{a^2}{2}$ or $x^2+y^2=\frac{a^2}{2}$.