Answer:
The equation of the parabola with the focus (-5, -1) and directrix y = -2 is
x^2 + 10x - 4y + 13 = 0
Explanation:
To find the equation of the parabola given the focus and directrix, we can use the geometric definition of a parabola.
The focus of the parabola is a point that is equidistant from the vertex and the directrix. Let's denote the vertex as (h, k).
The distance from the focus to the vertex is the same as the distance from the vertex to the directrix. In this case, the distance from the focus (-5, -1) to the directrix y = -2 is 1 unit.
Therefore, the vertex of the parabola is (h, k) = (-5, -3).
We can now write the equation of the parabola in vertex form, which is given by:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance from the vertex to the focus (or from the vertex to the directrix, since they are equidistant).
In our case, the vertex is (-5, -3) and the distance from the vertex to the directrix is 1 unit, so p = 1.
Substituting the values into the vertex form equation, we have:
(x - (-5))^2 = 4(1)(y - (-3))
Simplifying:
(x + 5)^2 = 4(y + 3)
Expanding:
x^2 + 10x + 25 = 4y + 12
Rearranging to get the equation in standard form:
x^2 + 10x - 4y + 13 = 0
Therefore,
The equation of the parabola is x^2 + 10x - 4y + 13 = 0.