Question

The locus of the mid points of the chords of the circle $x^2+y^2=4$ which subtend 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

Answer: (C)

Let mid point of the chord $AB$ is $C\left(x_1,y_1\right)$.
In $\Delta COB,\sin \frac{\pi }{4}=\frac{BC}{OB}$

$\Rightarrow \frac{1}{\sqrt{2}}=\frac{BC}{2}$
$\Rightarrow BC=\sqrt{2}$
Using Pythagoras theorem
$O{B}^2=O{C}^2+C{B}^2$
$\Rightarrow (2{)}^2=x_1^2+y_1^2+(\sqrt{2}{)}^2$
$\Rightarrow x_1^2+y_1^2=2$
Hence, locus of mid point of chord is
$x^2+y^2=2$