Find the equation of each of the following parabolas

Vertex at (0, 4), focus at (0, 2)

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

the distance between the vertex and directrix is same as the distance between the vertex and focus.

Directrix is y – 6 = 0

For any point of P(x, y) on the parabola

Distance of P from directrix = Distance of P from focus

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

Distance between the point of intersection & centre = [Distance Formula] {Between (x,y) & (0,2)}

Squaring both the sides,

x² + y² - 4y + 4 = (y – 6)²

x² + y² - 4y + 4 = y² - 12y + 36

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