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Question
Find the equation of the following parabolas:
Focus at (–1, –2), directrix x – 2y + 3 = 0
Solution
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.
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