To write the equation for a parabola with a focus at [tex]\((0, 5)\)[/tex] and a directrix at [tex]\(x = -7\)[/tex], follow these steps:
1. Determine the vertex of the parabola:
The vertex of a parabola is located midway between the focus and the directrix.
- The focus is at [tex]\((0, 5)\)[/tex].
- The directrix is the vertical line [tex]\(x = -7\)[/tex].
Since the vertex lies midway along the x-axis, we calculate:
[tex]\[
\text{vertex}_x = \frac{0 + (-7)}{2} = -3.5
\][/tex]
The y-coordinate of the vertex is the same as the y-coordinate of the focus, which is 5, so:
[tex]\[
\text{vertex}_y = \frac{5 + 5}{2} = 5
\][/tex]
Therefore, the vertex of the parabola is [tex]\((-3.5, 5)\)[/tex].
2. Identify the distance [tex]\(p\)[/tex]:
The distance [tex]\(p\)[/tex] is the distance from the vertex to the focus. Here, we have:
[tex]\[
p = |0 - (-3.5)| = 3.5
\][/tex]
3. Formulate the equation for the parabola:
For a parabola with a horizontal axis of symmetry (i.e., it opens left or right), the standard form is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex and [tex]\(p\)[/tex] is the focal distance. Substituting in the values we obtained:
[tex]\[
h = -3.5, \quad k = 5, \quad p = 3.5
\][/tex]
Thus, the equation of the parabola is:
[tex]\[
(y - 5)^2 = 4 \times 3.5 \times (x + 3.5)
\][/tex]
Simplifying:
[tex]\[
(y - 5)^2 = 14(x + 3.5)
\][/tex]
Therefore, the equation for the parabola is:
[tex]\[
(y - 5)^2 = 14(x + 3.5)
\][/tex]
