To determine the equation of the parabola with focus at [tex]\((-4, 5)\)[/tex] and directrix [tex]\(x = 16\)[/tex], we need to follow these steps:
1. Identify the Vertex:
The vertex of a parabola lies halfway between the focus and the directrix. Since the directrix is a vertical line at [tex]\(x = 16\)[/tex] and the focus is at [tex]\((-4, 5)\)[/tex], the x-coordinate of the vertex will be the midpoint of the x-coordinates of the focus and the directrix.
[tex]\[
\text{vertex}_x = \frac{\text{focus}_x + \text{directrix}_x}{2} = \frac{-4 + 16}{2} = \frac{12}{2} = 6
\][/tex]
The y-coordinate of the vertex remains the same as the y-coordinate of the focus because the directrix is vertical and does not affect the y-coordinate.
[tex]\[
\text{vertex}_y = 5
\][/tex]
Hence, the vertex is [tex]\((6, 5)\)[/tex].
2. Calculate [tex]\(p\)[/tex]:
[tex]\(p\)[/tex] is the distance from the vertex to the focus or to the directrix. This is calculated as the absolute difference between the x-coordinate of the vertex and the x-coordinate of the focus.
[tex]\[
p = \left|\text{vertex}_x - \text{focus}_x\right| = \left|6 - (-4)\right| = \left|6 + 4\right| = 10
\][/tex]
3. Form the Equation:
The standard form of the equation of a parabola that opens sideways (with a vertical directrix) is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
Here, the vertex is [tex]\((h, k)\)[/tex] and [tex]\(p\)[/tex] is the distance calculated in step 2.
Substituting [tex]\(h = 6\)[/tex], [tex]\(k = 5\)[/tex], and [tex]\(p = 10\)[/tex] into the equation:
[tex]\[
(x - 6)^2 = 4 \cdot 10 \cdot (y - 5)
\][/tex]
Simplifying this, we get:
[tex]\[
(x - 6)^2 = 40(y - 5)
\][/tex]
Thus, the equation of the parabola with focus [tex]\((-4, 5)\)[/tex] and directrix [tex]\(x = 16\)[/tex] is:
[tex]\[
(x - 6)^2 = 40(y - 5)
\][/tex]
