Question

Problem 3) Derive the equation of a parabola with the focus at (2,4) and the directrix at y=6. (1 point)
Responses

y=−14(x−2)2+5
y is equal to negative 1 fourth times open paren x minus 2 close paren squared plus 5

y=12(x+5)2+8
y is equal to 1 half times open paren x plus 5 close paren squared plus 8

y=2(x+5)2−1
y is equal to 2 times open paren x plus 5 close paren squared minus 1

y=9(x+1)2−7

Answer 1

To derive the equation of the parabola with the focus at \( (2, 4) \) and the directrix at \( y = 6 \), we first need to find the vertex of the parabola. The vertex \( V \) will be halfway between the focus and the directrix.

1. **Identify the coordinates of the focus and the directrix:**
- Focus: \( (2, 4) \)
- The directrix is the line \( y = 6 \).

2. **Find the vertex:**
The vertex lies halfway between the focus's y-coordinate and the directrix's y-coordinate:
- Vertex y-coordinate: \( \frac{4 + 6}{2} = 5 \)
- The x-coordinate of the vertex is the same as the focus's x-coordinate: \( 2 \).

So, the vertex \( V \) is at \( (2, 5) \).

3. **Determine the orientation of the parabola:**
Since the focus is below the directrix (focus at \( y = 4 \) and directrix at \( y = 6 \)), the parabola opens downwards.

4. **Use the standard form of a parabola:**
The general equation for a vertical parabola is given by:
\[
(x - h)^2 = 4p(y - k)
\]
where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus.

5. **Calculate \( p \):**
The distance \( p \) is the distance from the vertex \( (2, 5) \) to the focus \( (2, 4) \):
\[
p = 5 - 4 = 1
\]
Since the parabola opens downward, \( p \) will be negative:
\[
p = -1
\]

6. **Substituting values into the equation:**
- Vertex \( (h, k) = (2, 5) \)
- \( p = -1 \)

Thus, we substitute \( h \), \( k \), and \( p \) into the equation:
\[
(x - 2)^2 = 4(-1)(y - 5)
\]
Simplifying, we get:
\[
(x - 2)^2 = -4(y - 5)
\]
or:
\[
(x - 2)^2 = -4y + 20
\]
Rearranging for \( y \):
\[
y = -\frac{1}{4}(x - 2)^2 + 5
\]

7. **Final form of the equation:**
The equation of the parabola is:
\[
y = -\frac{1}{4}(x - 2)^2 + 5
\]

This matches the response option:
**y is equal to negative 1 fourth times open paren x minus 2 close paren squared plus 5**.