Question

The locus of the midpoints of chords of the circle $x^2+y^2=1$ which subtends a right angle at the origin is

• A. $x^2+y^2=\frac{1}{4}$
• B. $x^2+y^2=\frac{1}{2}$
• C. $xy=0$
• D. $x^2-y^2=0$

Answer 1

The correct option is B $x^2+y^2=\frac{1}{2}$

Let the centre of the circle $(0,0)$ be $C$. Therefore the length of $CM$ (considering the above given figure) is
$CM=R\sin \left(\frac{\pi }{4}\right)$ or $CM=\frac{R}{\sqrt{2}}$.
Hence
$C{M}^2=\frac{R^2}{2}$.
Applying distance formula:
$C{M}^2=(h-0{)}^2+(k-0{)}^2$
$=h^2+k^2$.
Hence
$C{M}^2=\frac{R^2}{2}$
or
$h^2+k^2=\frac{R^2}{2}$.
Replacing $(h,k)$ by $(x,y)$ and $R=1$, we get the equation of the midpoint of the chord as
$x^2+y^2=\frac{1}{2}$.