Question

Write the equation of the parabola with a focus of (6,−5) and directrix of x=2
𝑥
=
2
.



Note: Enter your answer and SHOW ALL THE STEPS NECESSARY to solve this problem in the space provided. You will not receive credit without all work shown.

Answer 1

To write the equation of a parabola given the focus and directrix, we can follow these steps:

1. **Identify the focus and directrix**:
- The focus is given as \((6, -5)\).
- The directrix is given as the line \(x = 2\).

2. **Determine the vertex of the parabola**:
- The vertex lies midway between the focus and the directrix.
- The focus \((6, -5)\) is 4 units away from the directrix line \(x = 2\) (specifically, \(6 - 2 = 4\)).
- The vertex's \(x\)-coordinate is the average of the focus's and directrix's locations:
\[
x_{vertex} = \frac{6 + 2}{2} = \frac{8}{2} = 4.
\]
- The \(y\)-coordinate of the vertex is the same as the \(y\)-coordinate of the focus (since the directrix is vertical, the \(y\)-coordinate doesn’t change):
\[
y_{vertex} = -5.
\]
- Thus, the vertex is at the point \((4, -5)\).

3. **Determine the orientation of the parabola**:
- Since the focus lies to the right of the directrix, the parabola opens to the right.

4. **Determine the value of \(p\)**:
- The value \(p\) is the distance from the vertex to the focus (or from the vertex to the directrix), which is \(p = 6 - 4 = 2\).
- Because the parabola opens to the right, \(p\) is positive.

5. **Write the standard form of the equation**:
- The standard form for a parabola that opens to the right is given by:
\[
(y - k)^2 = 4p(x - h)
\]
where \((h, k)\) is the vertex. In our case, \((h, k) = (4, -5)\) and \(p = 2\).

6. **Substitute the values into the equation**:
\[
(y - (-5))^2 = 4 \cdot 2 (x - 4)
\]
This simplifies to:
\[
(y + 5)^2 = 8(x - 4).
\]

Thus, the equation of the parabola is:
\[
\boxed{(y + 5)^2 = 8(x - 4)}.
\]