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# Derive the equation of the parabola with a focus at (-7, 5) and a directrix of y = -11

**Solution:**

Focus = (7, -5)

Directrix y = -11

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

Distance from focus point (7, -5) is √[(x − 7)^{2} + (y + 5)^{2}].

Distance from directrix y = -11 is |y + 11|

So the equation is

√[(x - 7)^{2} + (y + 5)^{2}] = |y + 11|

By squaring on both sides

(x - 7)^{2} + (y + 5)^{2} = (y + 11)^{2}

We get

x^{2} - 14x + 49 + y^{2} + 10y + 25 = y^{2} + 22y + 121

By further calculation

x^{2} - 14x - 47 - 11y = 0

x^{2} - 14x - 11y - 47 = 0

Therefore, the equation of the parabola is x^{2} - 14x - 11y - 47 = 0.

## Derive the equation of the parabola with a focus at (-7, 5) and a directrix of y = -11

**Summary:**

The equation of the parabola with a focus at (-7, 5) and a directrix of y = -11 is x^{2} - 14x - 11y - 47 = 0.

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