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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.
Solution
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.
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