Write the equation for a parabola with a focus at (-3,-5)(−3,−5) and a directrix at x=-7
Recall that the parabola
y^2 = 4px
has its
focus at (p,0)
directrix at x = -p
vertex is midway between the focus and the directrix, at (-5,-5)
So, here we clearly have p = 2
So the equations is
(y+5)^2 = 8(x+5)^2
see
https://www.wolframalpha.com/input/?i=parabola+%28y%2B5%29%5E2+%3D+8%28x%2B5%29
To write the equation for a parabola, we need to know its focus and directrix.
For a parabola with a focus at (h, k) and a directrix at x = a, the equation can be written in the form:
(x - h)^2 = 4a(y - k)
In this case, the focus is (-3, -5) and the directrix is x = -7.
The x-coordinate of the focus is h = -3, so we have (x + 3)^2.
The directrix is x = -7, so a = -7.
The y-coordinate of the focus is k = -5.
Putting it all together, the equation of the parabola is:
(x + 3)^2 = 4(-7)(y + 5)
Simplifying this equation, we get:
(x + 3)^2 = -28(y + 5)
And that is the equation for a parabola with a focus at (-3,-5) and a directrix at x=-7.