Question

The locus of the mid point 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

Answer: (C)

$x^2+y^2=2$

Given the equation of circle,

$x^2+y^2=4$

Let the centre of the circle be $O(0,0)$

and midpoint of the chord AB be $N(x,y)$

$\angle AOB=90°$ , since the chord subtends a right angle at origin.

$\angle NOB=\frac{90}{2}=45$

$\therefore$ Equation of circle is $x^2+y^2=4$

$\frac{ON}{OB}=\frac{\sqrt{x^2+y^2}}{2}$

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

By squaring both sides $\frac{1}{2}=\frac{x^2+y^2}{4}$

$⟹x^2+y^2=2$