To determine the equation of the parabola with a given vertex and focus, we need to understand the relationship between the vertex, focus, and the direction in which the parabola opens.
1. Identify the coordinates of the vertex and focus:
- Vertex, [tex]\( V(h, k) = (-7, -5) \)[/tex]
- Focus, [tex]\( F(f_1, f_2) = (-13, -5) \)[/tex]
2. Determine the orientation of the parabola:
Since the [tex]\(y\)[/tex]-coordinates for both the vertex and the focus are the same ([tex]\(-5\)[/tex]), the parabola opens either to the left or to the right, depending on the relative positions of the vertex and focus on the [tex]\(x\)[/tex]-axis.
3. Calculate the distance [tex]\(p\)[/tex]:
The distance [tex]\(p\)[/tex] represents the distance from the vertex to the focus:
[tex]\[
p = f_1 - h
\][/tex]
Plugging in the values from the coordinates:
[tex]\[
p = -13 - (-7) = -13 + 7 = -6
\][/tex]
The negative value indicates that the parabola opens to the left.
4. Determine the form of the equation of the parabola:
For a parabola that opens to the left or right, the general equation is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex and [tex]\( p \)[/tex] is the distance calculated in step 3.
5. Substitute the values into the standard form:
[tex]\[
(y - (-5))^2 = 4(-6)(x - (-7))
\][/tex]
Simplifying this:
[tex]\[
(y + 5)^2 = -24(x + 7)
\][/tex]
Hence, the equation of the parabola with vertex [tex]\((-7, -5)\)[/tex] and focus [tex]\((-13, -5)\)[/tex] is:
[tex]\[
(y + 5)^2 = -24(x + 7)
\][/tex]
