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Question
Find the equation of the parabola whose focus is the point F(-1, -2) and the directrix is the line 4x – 3y + 2 = 0.
Solution
F(-1, -2)
l : 4x – 3y + 2 = 0
Let P(x, y) be any point on the parabola.
FP = PM
⇒ FP2 = PM2
⇒ (x + 1)2 + (y + 2)2 = `[(4x - 3y + 2)/(sqrt(4^2 + (-3)^2))]^2`
⇒ x2 + 2x + 1 + y2 + 4y + 4 = `(16x^2 + 9y^2 + 4 - 24xy + 16x - 12y)/(16 + 9)`
⇒ 25(x2 + y2 + 2x + 4y + 5) = 16x2 + 9y2 – 24xy + 16x – 12y + 4
⇒ (25 – 16)x2 + (25 – 9)y2 + 24xy + (50 – 16)x + (100 + 12)y + 125 – 4 = 0
⇒ 9x2 + 16y2 + 24xy + 34x + 112y + 121 = 0
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