Question

Select the equation in conic form that represents a parabola with focus at (4,−1) and directrix
y
=
−
7
.

Equation #1:
(
x
−
4
)
2
=
3
(
y
+
1
)


Equation #2:
12
(
x
−
4
)
=
(
y
+
4
)
2


Equation #3:
(
x
−
4
)
2
=
12
(
y
+
4
)


(1 point)
Equation #
represents a parabola with the given focus and directrix.

Answer 1

To determine the correct equation of the parabola with the given focus at \((4, -1)\) and the directrix \(y = -7\), we can use the definition of a parabola. A parabola is the set of all points \((x, y)\) that are equidistant to the focus and the directrix.

1. **Finding the Vertex**: The vertex of the parabola is halfway between the focus and the directrix.
- The \(y\)-coordinate of the focus is \(-1\) and the \(y\)-coordinate of the directrix is \(-7\).
- The vertex \(y\)-coordinate is \((-1 + (-7)) / 2 = -4\). The \(x\)-coordinate is the same as that of the focus, which is \(4\). Therefore, the vertex is at \((4, -4)\).

2. **Finding \(p\)**: The distance \(p\) from the vertex to the focus (or from the vertex to the directrix) is found as follows:
- The distance from \(y = -4\) (vertex) to \(y = -1\) (focus) is \(3\), so \(p = 3\).

3. **Finding the Equation**: Since the focus is above the directrix, the parabola opens upwards. The standard form of a parabola that opens up is given by:
\[
(x - h)^2 = 4p(y - k)
\]
where \((h, k)\) is the vertex. Substituting our values:
- \(h = 4\)
- \(k = -4\)
- \(4p = 4 \times 3 = 12\)

Thus, the equation becomes:
\[
(x - 4)^2 = 12(y + 4)
\]

Let's compare this with the provided equations:

- **Equation #1**: \((x - 4)^2 = 3(y + 1)\) (not the correct p)
- **Equation #2**: \(12(x - 4) = (y + 4)^2\) (not the correct form)
- **Equation #3**: \((x - 4)^2 = 12(y + 4)\) (this is correct, but we'll rewrite it properly downwards)

Comparing all derived forms and provided forms, **the correct form is**:

**Equation #3: \((x - 4)^2 = 12 (y + 4)\)**

Thus, **Equation #3 represents a parabola with the given focus and directrix.**