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-
Text Solution
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If the centre of the circle is ( α , β ), then its equation is
(x – α )2 + (y – β )2 = a2 + ( β – k)2
Its intersection with the x-axis are given by
(x – α )2 + β 2 = α 2 + ( β – k)2
(i.e. x2 – 2 α x + (2 β k – 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
= 4 α 2 – 4 (2 β k – k2)
Locus of ( α , β ) is x2 – 2ky + k2 – 
= 0 which represents a parabola.
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