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# 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

x^{2} + y^{2} + 8y + 16 = y^{2} - 8y + 16

x^{2} + 16y = 0

Therefore, the equation of the parabola is x^{2} + 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 x^{2} + 16y = 0.

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