Derive the equation of the parabola with a focus at (0, -4) and a directrix of y = 4.

Solution:

Focus = (0, -4)

Directrix y = 4

Consider (x, y) as a point on the parabola

Distance from focus point (0, -4) is √(x − 0)² + (y + 4)².

Distance from directrix y = 4 is |y - 4|

So the equation is

√(x − 0)² + (y + 4)² = |y - 4|

By squaring on both sides

(x − 0)² + (y + 4)² = (y - 4)²

We get

x² + y² + 8y + 16 = y² - 8y + 16

x² + 16y = 0

Therefore, the equation of the parabola is x² + 16y = 0.

Derive the equation of the parabola with a focus at (0, -4) and a directrix of y = 4.

Summary:

The equation of the parabola with a focus at (0, -4) and a directrix of y = 4 is x² + 16y = 0.