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. [NCERT EXEMPLAR]

Answer 1

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

$$\begin{gathered} \Rightarrow \sqrt{{\left(x-0\right)}^2+{\left(y-2\right)}^2}=\frac{\left|y-6\right|}{1} \\ \Rightarrow x^2+y^2-4y+4=y^2-12y+36 \\ \Rightarrow x^2+8y=32 \end{gathered}$$