Question

The locus of the midpoints of a chord of a circle $x^2+y^2=4$, which subtends a right angle at the origin, is

• A. $x+y=2$
• B. $x^2+y^2=1$
• C. $x^2+y^2=2$
• D. $x+y=1$

Answer 1

The correct option is C $x^2+y^2=2$

Let mid point be $P(h,k)$
and origin be $O(0,0)$ which is also centre of the given circle.
Since chord making right angle at origin and P is mid point, OP will bisect the right angle.
$\Rightarrow$ OP $=r\cos {45}^0$ where $r$ is radius of the given circle.
$\Rightarrow OP=\sqrt{(h-0{)}^2+(k-0{)}^2}=2\times \frac{1}{\sqrt{2}}=\sqrt{2}$
Squaring we get, $h^2+k^2=2$
Hence locus of $P(h,k)$ is, $x^2+y^2=2$