Question

Derive the equation of a parabola with a focus at (6,−2) and a directrix at y=0.(1 point)
Responses

y=14x2+3x+8
y equals Start Fraction 1 over 4 End Fraction x squared plus 3 x plus 8

y=−14x2+3x−10
y equals negative Start Fraction 1 over 4 End Fraction x squared plus 3 x minus 10

y=14x2+x+4
y equals Start Fraction 1 over 4 End Fraction x squared plus x plus 4

y=−14x2+2x−6

Answer 1

To derive the equation of a parabola given its focus and directrix, we can start by noting the following:

- The focus of the parabola is at the point \( (6, -2) \).
- The directrix is the horizontal line \( y = 0 \).

### Step 1: Identify the vertex

The vertex of the parabola is located halfway between the focus and the directrix. Given the focus \( (6, -2) \) and the directrix \( y = 0 \):

- The \( y \)-coordinate of the vertex is the average of the \( y \) coordinate of the focus and the line of the directrix:
\[
y_v = \frac{-2 + 0}{2} = -1
\]

- The \( x \)-coordinate of the vertex is the same as the focus since the directrix is horizontal:
\[
x_v = 6
\]

Therefore, the vertex of the parabola is \( (6, -1) \).

### Step 2: Determine the distance \( p \)

The distance \( p \) from the vertex to the focus (or the vertex to the directrix) is:
\[
p = -2 - (-1) = -1
\]
(since the vertex is above the focus, \( p \) is negative, indicating the parabola opens downwards).

### Step 3: Use the parabola formula

The standard form of a parabola that opens vertically is:
\[
(x - h)^2 = 4p(y - k)
\]
where \( (h, k) \) is the vertex.

Substituting \( h = 6 \), \( k = -1 \) and \( p = -1 \) into the formula, we get:
\[
(x - 6)^2 = 4(-1)(y + 1)
\]

This simplifies to:
\[
(x - 6)^2 = -4(y + 1)
\]

### Step 4: Rearranging the equation

To express \( y \) in terms of \( x \), we can rearrange the equation:
\[
(y + 1) = -\frac{1}{4}(x - 6)^2
\]
\[
y = -\frac{1}{4}(x - 6)^2 - 1
\]

### Step 5: Expand the equation

Now we will expand \(-\frac{1}{4}(x - 6)^2 - 1\):
\[
(x - 6)^2 = x^2 - 12x + 36
\]
Thus,
\[
y = -\frac{1}{4}(x^2 - 12x + 36) - 1
\]
\[
y = -\frac{1}{4}x^2 + 3x - 9 - 1
\]
\[
y = -\frac{1}{4}x^2 + 3x - 10
\]

### Conclusion

The equation of the parabola is:
\[
y = -\frac{1}{4}x^2 + 3x - 10
\]

From the provided responses, the correct answer is:
\[
\text{y = } -\frac{1}{4}x^2 + 3x - 10
\]