Question

Find the equation of the following parabolas:

Vertex at (0, 4), focus at (0, 2)

Answer 1

Given that vertex at (0, 4) and focus at (0, 2).

So, the equation of directrix is y – 6 = 0

According to the definition of the parabola

PF = PM.

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

⇒ `sqrt(x^2 + y^2 + 4 - 4y) = |y - 6|`

Squaring both the sides, we get

x² + y² + 4 – 4y = y² + 36 – 12y

⇒ x² + 4 – 4y = 36 – 12y

⇒ x² + 8y – 32 = 0

⇒ x² = 32 – 8y

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