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 $P(h,k)$ is the mid point of chord $AB$


$△OPA$ is the right angle triangle
$\angle POB=\frac{{90}^{\circ }}{2}={45}^{\circ }$
$\therefore \cos {45}^{\circ }=\frac{OP}{OB}=\frac{OP}{1}$
$\Rightarrow OP=\frac{1}{\sqrt{2}}$
By distance formula, $OP=\sqrt{(h-0{)}^2+(k-0{)}^2}$
$\Rightarrow \sqrt{h^2+k^2}=\frac{1}{\sqrt{2}}$
$\Rightarrow h^2+k^2=\frac{1}{2}$
Put $(x,y)$ at the place of $(h,k)$ to get the locus
So,
$x^2+y^2=\frac{1}{2}$