Question

From the origin, chords are drawn to the circle $x^2+y^2$ -2y $=0$. The locus of the middle point of these chords is

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

Answer 1

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

Let $e{q}^n$ of chord passing through origin is $y=mx$

$x^2+y^2-2y=0$

$x^2+m^2{x}^2-2mx=0$

$x=0,x=\frac{2m}{1+m^2}$

then $x=0,y=0$

and $x=\frac{2m}{1+m^2},y=\frac{2m^2}{1+m^2}$

$(\frac{2m}{1+m^2},\frac{2m^2}{1+m^2})$

mid-point of AB is

$x=\frac{m}{1+m^2}andy=\frac{m^2}{1+m^2}$

$\therefore x^2+y^2=\frac{m^2}{1+m^2}=y$

$\Rightarrow x^2+y^2-y=0$