a parabola has an equation of a directrix l=x=2, determine the standard equation if the focus is 8 units away from the directrix

Given that the directrix of the parabola is \( x = 2 \), and the focus is 8 units away from the directrix, the parabola is oriented horizontally.

### Step 1: Determine the focus coordinates
Since the directrix is \( x = 2 \) and the focus is 8 units away from the directrix, the distance between the directrix and the focus is 8 units. 

- If the parabola opens to the right, the focus will be at \( x = 2 + 8 = 10 \).
- If the parabola opens to the left, the focus will be at \( x = 2 - 8 = -6 \).

Let's consider the general case where the focus is at \( (h, k) \).

### Step 2: General formula for the parabola
The standard form of the equation for a parabola that opens horizontally is:
\[
(y - k)^2 = 4p(x - h)
\]
where:
- \( (h, k) \) is the focus of the parabola,
- \( p \) is the distance from the vertex to the focus.

### Step 3: Determine the equation
The vertex \( V \) is halfway between the directrix and the focus. Therefore, the vertex \( V \) lies at:
- If the focus is at \( (10, k) \), the vertex is \( V = \left(\frac{2+10}{2}, k\right) = (6, k) \).
- If the focus is at \( (-6, k) \), the vertex is \( V = \left(\frac{2-6}{2}, k\right) = (-2, k) \).

The value of \( p \) (the distance from the vertex to the focus) is \( p = 4 \).

Thus, the possible equations for the parabola are:
1. If the focus is at \( (10, k) \), the equation is:
   \[
   (y - k)^2 = 16(x - 6)
   \]
2. If the focus is at \( (-6, k) \), the equation is:
   \[
   (y - k)^2 = -16(x + 2)
   \]

### Final Answer:
- If the parabola opens to the right: \((y - k)^2 = 16(x - 6)\)
- If the parabola opens to the left: \((y - k)^2 = -16(x + 2)\)

Would you like to know the next steps for finding the specific equation based on additional conditions, or do you have any questions? 

Here are 8 related questions:
1. How do you determine the direction a parabola opens based on the focus and directrix?
2. What is the significance of the distance \( p \) in the equation of a parabola?
3. Can you explain how to find the vertex given the focus and directrix?
4. How does the equation of a parabola change if the directrix is vertical vs. horizontal?
5. What happens to the parabola if the focus is moved further away from the directrix?
6. How do you derive the equation of a parabola given the vertex and focus?
7. What are the properties of a parabola's axis of symmetry?
8. How do you find the length of the latus rectum for a given parabola?

**Tip:** For any parabola, the distance between the vertex and the focus (denoted as \( p \)) is always equal to the distance from the vertex to the directrix.

Learn how to determine the standard equation of a parabola when given the focus and the directrix. This problem involves understanding parabolas in coordinate geometry, focusing on the relationship between the focus, directrix, and vertex. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation