Question

If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.

Answer 1

Given that: Vertex = (0, 4) and Focus = (0, 2)

Let P(x, y) be any point on the parabola.

PB is perpendicular to the directrix.

We have PF = PB

⇒ `sqrt((x - 0)^2 + (y - 2)^2) = |(0 + y - 6)/sqrt(0 + 1)|`

⇒ `sqrt(x^2 + (y - 2)^2) = (y - 6)`   .......[Equation of directrix is y = 6]

Squaring both sides, we have

x² + (y – 2)² = (y – 6)²

⇒ x² + y² + 4 – 4y = y² + 36 – 12y

⇒ x² – 4y + 12y – 32 = 0

⇒ x² + 8y – 32 = 0

Hence, the required equation is x² + 8y = 32.