Question

The locus of the centre of a circle which passes through the origin and cuts-off a length $2$b from the line $x=c$ is?

• A. $y^2+2cx=b^2+c^2$
• B. $x^2+cx=b^2+c^2$
• C. $y^2+2xy=b^2+c^2$
• D. $x^2+cy=b^2+c^2$

Answer 1

The correct option is A $y^2+2cx=b^2+c^2$

The length of the intercept cut by the circle - $2\sqrt{r^2-d^2}$,where $d$ is the perpendicular distance between centre and the line.

Hence, $2b=2\sqrt{r^2-d^2}$

$b=\sqrt{r^2-d^2}$

Let the centre of the circle $C(x,y)$

$d=x-c$

$r=\sqrt{x^2+y^2}$

$b=\sqrt{x^2+y^2-(x-c{)}^2}$

$b^2=y^2+2cx-c^2$

$b^2+c^2=y^2+2cx$