Question

The locus of the middle points of those chords of the circle $x^2+y^2=4$ which subtend a right angle at the origin is

• A. $x^2+y^2-2x-2y=0$
• B. $x^2+y^2=2$
• C. $x^2+y^2=4$
• D. $(x-1{)}^2+(y-2{)}^2=5$

Answer 1

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

Equation of circle-

$x^2+y^2=4$

Centre of circle $\equiv (0,0)$

radius of circle $=2$

Let midpoint of the chord $AB$ be $M(x,y)$

Distance of $M(x,y)$ from the origin $=\sqrt{x^2+y^2}$

$\angle AOB=90°[\because \text{chord subtends a right angle at the origin}]$

$\angle MOB=\frac{\angle AOB}{2}=45°$

Now in $△MOB$

$\because △MOB$ is a right angle triangle.

$\therefore \cos \theta =\frac{ON}{OB}$

$\Rightarrow \cos 45=\frac{\sqrt{x^2+y^2}}{2}[\because \text{OB is radius of circle}]$

$\Rightarrow \frac{1}{\sqrt{2}}=\frac{\sqrt{x^2+y^2}}{2}$

$\sqrt{x^2+y^2}=\sqrt{2}$

Squaring both sides, we get

$x^2+y^2=2$

Hence the correct answer is $x^2+y^2=2$.