Question

Write the equation of the parabola with a focus at (10,−3) and a directrix of x=2. It will be helpful to draw a sketch first and identify the information needed to write the equation.(

Answer 1

To write the equation of the parabola, we need to analyze the information given regarding the focus and directrix.

1. **Identify the Characteristics of the Parabola:**
- The focus of the parabola is located at \( (10, -3) \).
- The directrix is a vertical line defined by the equation \( x = 2 \).
- Since the focus is to the right of the directrix, the parabola opens to the right.

2. **Determine the Vertex:**
- The vertex of the parabola is located halfway between the focus and the directrix.
- The x-coordinate of the vertex can be determined by averaging the x-coordinates of the focus and the line \( x = 2 \):
\[
\text{Vertex (h)} = \frac{10 + 2}{2} = \frac{12}{2} = 6.
\]
- The y-coordinate of the vertex is the same as the y-coordinate of the focus since the directrix is vertical. Therefore, the y-coordinate of the vertex is \( -3 \).

So, the vertex is at \( (6, -3) \).

3. **Determine the Distance \( p \):**
- The distance \( p \) is the distance from the vertex to the focus (or to the directrix).
- Since the focus is at \( (10, -3) \) and the vertex is at \( (6, -3) \),
\[
p = 10 - 6 = 4.
\]
- This means the parabola opens to the right and \( p = 4 \).

4. **Write the Equation of the Parabola:**
- The general form of the equation of a horizontally opening parabola with vertex at \( (h, k) \) is given by:
\[
(y - k)^2 = 4p(x - h).
\]
- Substituting \( h = 6 \), \( k = -3 \), and \( p = 4 \):
\[
(y + 3)^2 = 16(x - 6).
\]

Thus, the equation of the parabola is:
\[
(y + 3)^2 = 16(x - 6).
\]