# 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

x^{2} + 10x + 25 + y^{2} - 10y + 25 = y^{2} + 2y + 1

By further calculation

x^{2} + y^{2} + 10x - 10 y + 50 = y^{2} + 2y + 1

Grouping the similar terms

x^{2} + y^{2} + 10x - 10 y + 50 - y^{2} - 2y - 1 = 0

x^{2} + 10x - 12y + 49 = 0

Therefore, the equation of the parabola is x^{2} + 10x - 12y + 49 = 0.

## 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 x^{2} + 10x - 12y + 49 = 0.

Math worksheets and

visual curriculum

visual curriculum