Question

A circle touches x-axis and cuts off a constant length 2l from the y-axis. The locus of its centre is

• A. $x^2$ + $y^2$ = $l^2$
• B. $x^2$ + $y^2$ = 2$l^2$
• C. $x^2$ + $y^2$ = 3$l^2$
• D. $x^2$ - $y^2$ = - $l^2$

Answer 1

The correct option is D $x^2$ - $y^2$ = - $l^2$

Let C(h, k) be the centre of the circle
Circle touches x-axis $\Rightarrow$ radius = $\left|k\right|$
Equation of the circle with centre (h, k) and radius $\left|k\right|$ is
$(x-h{)}^2+(y-k{)}^2=k^2$ $\Rightarrow$ $x^2$ + $y^2-2hx-2ky+h^2$ = 0
Length of the intercept on y-axis is 2l $\Rightarrow$ 2$\sqrt{k^2-h^2}$ = 2l $\Rightarrow k^2-h^2=l^2$
Locus of (h, k) is $y^2$ - $x^2$ = $l^2$