Derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7
Solution:
The focus of the parabola is (-5, -5) and a directrix is y = 7
The standard equation of a parabola is
(x - h)2 = 4p(y - k)
Where y = k - p is the directrix
(h, k + p) is the focus
(h, k + p) = (-5, -5)
By comparison
h = -5
k + p = -5 --- (1)
As directrix is y = 7
k + p = 7 --- (2)
Now solve the linear equations. Adding (1) and (2), we get
2k = 2
k = 1
Substitute the value of k in (1)
k + p = - 5
1 + p = -5
p = - 5 - 1 = - 6
Substitute p = -6, h = -5 and k = 1 in the standard equation
(x + 5)2 = 4(-6)(y - 1)
(x + 5)2 = -24(y - 1)
Therefore, the equation of the parabola is (x + 5)2 = -24(y - 1).
Aliter:
Given that, Focus = (-5, -5) and directrix y = 7
Let us suppose that there is a point (x, y) on the parabola.
Its distance from the focus point (-5, -5) is √(x + 5)2 + (y + 5)2
Its distance from directrix y = 7 is |y - 7|
Therefore, the equation will be:
√(x + 5)2 + (y + 5)2 = |y - 7|
Squaring on both sides,
(x + 5)2 + (y + 5)2 = (y - 7)2
x2 + 10x + 25 + y2 + 10y + 25 = y2 - 14y + 49
x2 + 10x + 24y + 1 = 0
Derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7
Summary:
The equation of the parabola with a focus at (-5, -5) and a directrix of y = 7 is (x + 5)2 = -24(y - 1).