If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.

Vertex = (0,4) Focus = (0,2)

So, the directrix of the parabola is y = 6,

Since, Distance of (x, y) from (0, 2) and perpendicular distance from (x, y) to directrix are always equal.

Using Distance Formula & Perpendicular Distance Formula,

Perpendicular Distance (Between a point and line) = , whereas the point is and the line is expressed as ax + by + c = 0 i.e.., PM x(0) + y – 6 = 0 & (x, y)

Distance between the point of intersection & centre = [Distance Formula]

Squaring both the sides,

x² + y² - 4y + 4 = y² - 12y + 36

x² + 8y – 32 = 0

Hence, the required equation is x² + 8y – 32 = 0.