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Question
If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.
Solution
Given that: Vertex = (0, 4) and Focus = (0, 2)
Let P(x, y) be any point on the parabola.
PB is perpendicular to the directrix.
We have PF = PB
⇒ `sqrt((x - 0)^2 + (y - 2)^2) = |(0 + y - 6)/sqrt(0 + 1)|`
⇒ `sqrt(x^2 + (y - 2)^2) = (y - 6)` .......[Equation of directrix is y = 6]
Squaring both sides, we have
x2 + (y – 2)2 = (y – 6)2
⇒ x2 + y2 + 4 – 4y = y2 + 36 – 12y
⇒ x2 – 4y + 12y – 32 = 0
⇒ x2 + 8y – 32 = 0
Hence, the required equation is x2 + 8y = 32.
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