Question

A circle whose centre is (4, –1) passes through the focus of the parabola x² + 16y = 0.

Show that the circle touches the directrix of the parabola.

Answer 1

Given parabola is x² = – 16y.

Comparing with x² = – 4by, we get,

4b = 16

∴ b = 4

∴ focus S = (0, – b) = (0, – 4)

The equation of the directrix is

y – b = 0

∴  y – 4 = 0

Let r be the radius of the circle drawn with centre C = (4, –1)

∵ S lies on the circle

∴ r = l(CS)

= `sqrt((4 - 0)^2 + (-1 + 4)^2`

= `sqrt(16 + 9)`

= `sqrt(25)`

= 5 units

The perpendicular distance of C = (4, –1) from the directrix i.e. from y – 4 = 0 is `|(-1 - 4)/sqrt(0 + 1)|` = 5 = r

∴ the circle touches the directrix of the parabola.