Derive the equation of the parabola with a focus at (-5, 5) and a directrix of y = -1.

Solution:

Focus = (-5, 5)

Directrix y = -1

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

Distance from focus point (-5, 5) is √[(x + 5)² + (y - 5)²]

Distance from directrix y = -1 is |y + 1|

So the equation is

√[(x + 5)² + (y - 5)²] = |y + 1|

By squaring on both sides

(x + 5)² + (y - 5)² = (y + 1)²

We get

x² + 10x + 25 + y² - 10y + 25 = y² + 2y + 1

By further calculation

x² + y² + 10x - 10 y + 50 = y² + 2y + 1

Grouping the similar terms

x² + y² + 10x - 10 y + 50 - y² - 2y - 1 = 0

x² + 10x - 12y + 49 = 0

Therefore, the equation of the parabola is x² + 10x - 12y + 49 = 0.

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Derive the equation of the parabola with a focus at (-5, 5) and a directrix of y = -1.

Summary:

The equation of the parabola with a focus at (-5, 5) and a directrix of y = -1 is x² + 10x - 12y + 49 = 0.