Question

Derive the equation of a parabola with a focus at  (6,−2)  and a directrix at  y=0 .(1 point)Responsesy=14x2+x+4y equals Start Fraction 1 over 4 End Fraction x squared plus x plus 4y=−14x2+2x−6y equals negative Start Fraction 1 over 4 End Fraction x squared plus 2 x minus 6y=−14x2+3x−10y equals negative Start Fraction 1 over 4 End Fraction x squared plus 3 x minus 10y=14x2+3x+8y equals Start Fraction 1 over 4 End Fraction x squared plus 3 x plus 8

Answer 1

To derive the equation of a parabola given a focus and a directrix, we can use the definition of a parabola as the set of all points equidistant from the focus and the directrix.

1. **Identify the coordinates of the focus and the equation of the directrix:**

- Focus: \( (6, -2) \)
- Directrix: \( y = 0 \)

2. **Let \( (x, y) \) be a point on the parabola.** The distance from this point to the focus is:

\[
\sqrt{(x - 6)^2 + (y + 2)^2}
\]

The distance from this point to the directrix \( y = 0 \) is simply the vertical distance:

\[
|y - 0| = |y|
\]

3. **Set these distances equal to each other** since points on the parabola are equidistant from the focus and the directrix:

\[
\sqrt{(x - 6)^2 + (y + 2)^2} = |y|
\]

4. **Square both sides to eliminate the square root:**

\[
(x - 6)^2 + (y + 2)^2 = y^2
\]

5. **Expand the equation:**

\[
(x - 6)^2 + y^2 + 4y + 4 = y^2
\]

The \( y^2 \) terms cancel out:

\[
(x - 6)^2 + 4y + 4 = 0
\]

6. **Rearranging gives:**

\[
4y = -(x - 6)^2 - 4
\]

Dividing by 4:

\[
y = -\frac{1}{4}(x - 6)^2 - 1
\]

7. **We can write it in a more standard form. Expanding gives:**

\[
y = -\frac{1}{4}(x^2 - 12x + 36) - 1
\]

Distributing the \(-\frac{1}{4}\):

\[
y = -\frac{1}{4}x^2 + 3x - 9 - 1
\]

Simplifying:

\[
y = -\frac{1}{4}x^2 + 3x - 10
\]

So, the equation of the parabola is:

\[
y = -\frac{1}{4}x^2 + 3x - 10
\]

From the provided response options, the correct one is:

\[
y = -\frac{1}{4} x^2 + 3x - 10
\]