# Derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7

**Solution:**

The focus of the parabola is (-5, -5) and a directrix is y = 7

The standard equation of a parabola is

(x - h)^{2} = 4p(y - k)

Where y = k - p is the directrix

(h, k + p) is the focus

(h, k + p) = (-5, -5)

By comparison

h = -5

k + p = -5 --- (1)

As directrix is y = 7

k + p = 7 --- (2)

Now solve the linear equations. Adding (1) and (2), we get

2k = 2

k = 1

Substitute the value of k in (1)

k + p = - 5

1 + p = -5

p = - 5 - 1 = - 6

Substitute p = -6, h = -5 and k = 1 in the standard equation

(x + 5)^{2} = 4(-6)(y - 1)

(x + 5)^{2} = -24(y - 1)

Therefore, the equation of the parabola is (x + 5)^{2} = -24(y - 1).

Aliter:

Given that, Focus = (-5, -5) and directrix y = 7

Let us suppose that there is a point (x, y) on the parabola.

Its distance from the focus point (-5, -5) is √(x + 5)^{2} + (y + 5)^{2}

Its distance from directrix y = 7 is |y - 7|

Therefore, the equation will be:

√(x + 5)^{2} + (y + 5)^{2} = |y - 7|

Squaring on both sides,

(x + 5)^{2} + (y + 5)^{2} = (y - 7)^{2}

x^{2} + 10x + 25 + y^{2} + 10y + 25 = y^{2} - 14y + 49

x^{2} + 10x + 24y + 1 = 0

## Derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7

**Summary:**

The equation of the parabola with a focus at (-5, -5) and a directrix of y = 7 is (x + 5)^{2} = -24(y - 1).

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