Question

The locus of the mid-point of the chord of a circle $x^2+y^2=4$ such that the segment intercepted by the chord on the curve $x^2-2x–2y=0$ subtends a right angle at the origin 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. $x^2+y^2-2x+2y=0$

Answer 1

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

Let mid-point of chord be (h, k)
Equation of chord
$xh+ky=h^2+k^2$
$x^2-2x-2y=0$
Homogenising,
$x^2-2x\left(\frac{xh+ky}{h^2+k^2}\right)-2y\left(\frac{xh+ky}{h^2+k^2}\right)=0$
Angle subtended at origin is ${90}^{\circ }$
$\therefore 1-\frac{2h}{h^2+k^2}-\frac{2k}{h^2+k^2}=0$
$\Rightarrow h^2+k^2-2h-2k=0$
$\therefore$ Required locus
$x^2+y^2-2x-2y=0$