Question

The equation of the parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is ______.

Answer 1

The equation of the parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is 4x² + 4xy + y² + 4x + 32y + 16 = 0.

Explanation:

. Let (x₁, y₁) be any point on the parabola.

According to the definition of the parabola

`sqrt((x_1 + 1)^2 + (y_1 + 2)^2) = |(x_1 - 2y_1 + 3)/sqrt((1)^2 + (-2)^2)|`

Squaring both sides, we get

`x_1^2 + 1 + 2x_1 + y_1^2 + 4 + 4y_1 = (x_1^2 + 4y_1^2 + 9 - 4x_1y_1 - 12y_1 + 6x_1)/5`

⇒ `x_1^2 + y_1^2 + 2x_1 + 4y_1 + 5 = (x_1^2 + 4y_1^2 - 4x_1y_1 - 12y_1 + 6x_1 + 9)/5`

⇒ `5x_1^2 + 5y_1^2 + 2x_1 + 10x_1 + 20y_1 + 25 = x_1^2 + 4y_1^2 - 4x_1y_1 - 12y_1 + 6x_1 + 9`

⇒ `4x_1^2 + y_1^2 + 4x_1 + 32y_1 + 4x_1y_1 + 16` = 0