Question

the locus of a midpoint of a chord of the 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 the end points of chord be $A(2cos\theta ,2sin\theta )$ and $B(2cos\alpha ,2sin\alpha )$

Let the midpoint be $(h,k)=(cos\theta +cos\alpha ,sin\theta +sin\alpha )$

Given that $OA$ and $OB$ are perpendicular

Slope of $OA$ is $tan\theta$ and slope of $OB$ is $tan\alpha$

So we have $tan\theta tan\alpha =-1$

$\Rightarrow tan(\theta -\alpha )=\infty$

$\Rightarrow tan(\theta -\alpha )=0$

We have $h^2+k^2=(cos\theta +cos\alpha {)}^2+(sin\theta +sin\alpha {)}^2$

$\Rightarrow h^2+k^2=1+1+cos(\theta -\alpha )=2$

Therefore the correct option is $C$