Question

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

• A. A circle
• B. A parabola
• C. An ellipse
• D. A hyperbola

Answer 1

Answer: (B)

A parabola

Answer 2

If the centre of the circle is (a, b), then its equation is

(x – a)2 + (y – b)2 = a2 + (b – k)2

Its intersection with the x-axis are given by

(x ­– a)2 + b2 = a2 + (b ­– k)2

(i.e. x2 – 2ax + (2bk – k2) = 0

Its roots x1, x2 are the x coordinates of A

and B. Given |x2 – x1| = constant c. we have

c2 = (x2 – x1)2 = (x2 + x1)2 – 4x2x1

= 4a2 – 4 (2bk – k2)

Locus of (a, b) is x2 – 2ky + k2 – $\frac{c^{2}}{4}$ = 0 which represents a parabola.