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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
As the vertex and focus lie on y-axis, so y-axis is the axis of the parabola.
If the directrix meets the axis of the parabola at point Z, the AZ = AF = 2
OZ = OF + AZ + FA = 2 + 2 + 2 = 6
So, the equation of the directrix is y = 6
i.e., y − 6 = 0
Let P(x, y) be any point in the plane of the focus and directrix and MP be the perpendicular
distance from P to the directrix, then P lies on parabola iff FP = MP
\[\Rightarrow \sqrt{\left( x - 0 \right)^2 + \left( y - 2 \right)^2} = \frac{\left| y - 6 \right|}{1}\]
\[ \Rightarrow x^2 + y^2 - 4y + 4 = y^2 - 12y + 36\]
\[ \Rightarrow x^2 + 8y = 32\]
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