Write the equation for a parabola with a focus at (-4,3) and a directrix at y = 5.

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The equation for a parabola with a focus at (-4, 3) and a directrix at y = 5 is; y = ¹/₄(x + 4)² + 4

The standard form of the equation of a parabola is given by;

(x - h)² = 4p(y - k)

where;

We are given the coordinates of the focus of the parabola as (-4, 3)

Thus, from (h, k + p)

h = -4

k + p = 3   (eq 1)

From y = k - p,  since it has a directrix of y = 5, then;

k - p = 5   (eq 2)

→ Add equations (1) to (2) to get;

2k = 8

k = 8/2

k = 4

Put 4 for k in equation (1) to get

4 + p = 3

p = 4 - 3

p = 1

Plugging in the relevant values into the standard form of the equation of the parabola gives (x - h)² = 4p(y - k)

(x - (-4))² = 4(1)(y - 4)

⇒ (x + 4)² = 4(y - 4)

⇒ y = ¹/₄(x + 4)² + 4

The equation of the parabola is y = ¹/₄(x + 4)² + 4

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