Question

The locus of the mid points of the chords of the circle $x^2+y^2+2x-2y-2=0$ which makes an angle of $90°$ at the centre is

• A. $x^2+y^2-2x-2y=0$
• B. $x^2+y^2-2x+2y=0$
• C. $x^2+y^2+2x-2y=0$
• D. $x^2+y^2+2x-2y-1=0$

Answer 1

The correct option is C $x^2+y^2+2x-2y=0$

Equation of circle $(x+1{)}^2+(y-1{)}^2=2^2$
Let the end point of chords
$(x_1,y_1)=(2\cos \theta -1,2\sin \theta +1)$
$(x_2,y_2)=(2\sin \theta -1,-2\cos \theta +1)$
Mid point of chord $M=(\frac{2\cos \theta -1+2\sin \theta -1}{2},\frac{2\sin \theta +1-2\cos \theta +1}{2})$
$M=(\cos \theta +\sin \theta -1,\sin \theta -\cos \theta +1)$
$\Rightarrow (h,k)=(\cos \theta +\sin \theta -1,\sin \theta -\cos \theta +1)$
Locus of mid point
$(h+1{)}^2+(k-1{)}^2=2\Rightarrow h^2+k^2+2h-2k=0$
$\Rightarrow x^2+y^2+2x-2y=0$