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

x² - 14x + 49 + y² + 10y + 25 = y² + 22y + 121

By further calculation

x² - 14x - 47 - 11y = 0

x² - 14x - 11y - 47 = 0

Therefore, the equation of the parabola is x² - 14x - 11y - 47 = 0.

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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 x² - 14x - 11y - 47 = 0.