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