Question

From the origin chords are drawn to the circle ${(x-1)}^2+y^2=1$.The locus of the midpoints of these chords is

• A. $x^2+y^2+x=0$
• B. $x^2+y^2-x=0$
• C. $x^2-y^2-x=0$
• D. none of these

Answer 1

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

The circle is ${(x-1)}^2+y^2=1$
$\Rightarrow x^2+y^2-2x=0$
The chord with midpoint $P(x_1,y_1)$ is $S_1=S_{11}$
$\Rightarrow x_{1}x+y_{1}y-(x+x_1)=x_1^2+y_1^2-2x_1$
It passes through $(0,0)$
$\therefore x_1^2+y_1^2-x_1=0$
The locus of $P$ is $x^2+y^2-x=0$