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)

Answer 1

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)