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.
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.