Question

Find the equation of the following parabolas:

Focus at (–1, –2), directrix x – 2y + 3 = 0

Answer 1

Given that focus at (– 1, – 2) and directrix x – 2y + 3 = 0

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

According to the definition of the parabola, we have

PF = PM

`sqrt((x + 1)^2 + (y + 2)^2) = |(x - 2y + 3)/sqrt((1)^2 - (-2)^2)|`

⇒ `sqrt(x^2 + 1 + 2x + y^2 + 4 + 4y) = |(x - 2y + 3)/sqrt(5)|`

Squaring both sides, we get

x² + 1 + 2x + y² + 4 + 4y = `(x^2 + 4y^2 + 9 - 4xy - 12y + 6x)/5`

⇒ 5x² + 5 + 10x + 5y² + 20 + 20y = x² + 4y² + 9 – 4xy – 12y + 6x

⇒ 4x² + y² + 4xy + 4x + 32y + 16 = 0

Hence, the required equation is 4x² + 4xy + y² + 4x + 32y + 16 = 0.