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

Question

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.