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Question
If the vertex of the parabola is the point (–3, 0) and the directrix is the line x + 5 = 0, then its equation is ______.
Options
y2 = 8(x + 3)
x2 = 8(y + 3)
y2 = – 8(x + 3)
y2 = 8(x + 5)
Solution
If the vertex of the parabola is the point (–3, 0) and the directrix is the line x + 5 = 0, then its equation is y2 = 8(x + 3).
Explanation:
Given that vertex = (– 3, 0)
∴ a = – 3
And directrix is x + 5 = 0
We get AF = AD
i.e., A is the mid-point of DF
∴ `3 = (x_1 - 5)/2`
⇒ `x_1 = -6 + 5` = – 1
And 0 = `(0 + y_1)/2`
⇒ `y_1 = 0`
∴ Focus F = (– 1, 0)
Now `sqrt((x + 1)^2 + (y - 0)^2) = |(x + 5)/sqrt(1^2 + 0^2)|`
Squaring both sides, we get
(x + 1)2 + y2 = (x + 5)2
⇒ x2 + 1 + 2x + y2 = x2 + 25 + 10x
⇒ y2 = 10x – 2x + 24
⇒ y2 = 8x + 24
⇒ y2 = 8(x + 3)
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