The locus of the centre of circle which cuts off an intercept of constant length on the x-axis and which passes through a fixed point on the y-axis, is

Options

A. a circle
B. a parabola
C. an ellipse
D. a hyperbola

Answer

B

Explanation

Let the circle be `x^(2)+y^(2)+2gx+2fy+c=0`.
Suppose it cuts off an intercept of length 2l on x-axis. Then,
`2sqrt(g^(2)-c) = al rArr g^(2)-c=l^(2) " " ...(i)`
The circle passes through a fixed point `(0, lambda)` on y-axis.
`lambda^(2)+2f lambda +c=0 " " ...(ii)`
Eliminating c from (i) and (ii), we get `g^(2)+2f lambda + lambda^(2)=l^(2)`
Hence, the locus of `(-g, -f) ` is `x^(2)-2lambda y + lambda^(2)=0`.
Clearly, it is a parabola.