Determine the equation of a parabola with focus at focus (4, 2) and directrix of y = -4.

To determine the equation of a parabola given its focus and directrix, we can use the definition of a parabola. A parabola is the set of all points \((x, y)\) such that the distance from any point \((x, y)\) on the parabola to the focus is equal to the distance from that point to the directrix.

### Given:
- Focus: \((4, 2)\)
- Directrix: \(y = -4\)

### Step 1: Find the vertex
The vertex of the parabola is equidistant from the focus and the directrix. The distance between the focus and the directrix is:

\[
\text{Distance} = \text{Focus y-coordinate} - \text{Directrix y-coordinate} = 2 - (-4) = 6
\]

Since the vertex is halfway between the focus and the directrix, its y-coordinate will be:

\[
\text{Vertex y-coordinate} = -4 + \frac{6}{2} = -4 + 3 = -1
\]

So, the vertex is at \((4, -1)\).

### Step 2: Determine the equation of the parabola
The standard form of a parabola that opens vertically (up or down) is:

\[
(x - h)^2 = 4p(y - k)
\]

Where:
- \((h, k)\) is the vertex
- \(p\) is the distance from the vertex to the focus

Here, the vertex is \((4, -1)\) and \(p\) is the distance from the vertex to the focus:

\[
p = \text{Focus y-coordinate} - \text{Vertex y-coordinate} = 2 - (-1) = 3
\]

Since the focus is above the vertex, the parabola opens upwards, so \(p\) is positive.

### Final Equation:
Substituting \(h = 4\), \(k = -1\), and \(p = 3\) into the equation:

\[
(x - 4)^2 = 4(3)(y + 1)
\]

Simplifying:

\[
(x - 4)^2 = 12(y + 1)
\]

This is the equation of the parabola with focus \((4, 2)\) and directrix \(y = -4\).

Would you like further details on this solution, or do you have any questions?

### Related Questions:
1. How would the equation change if the directrix was a vertical line instead of horizontal?
2. What would the equation be if the parabola opened downward?
3. How do we find the equation of a parabola given its vertex and one point on the parabola?
4. Can you derive the general form of the equation of a parabola from the standard form?
5. What is the role of the parameter \(p\) in determining the shape of the parabola?

### Tip:
Always remember that the sign of \(p\) determines the direction in which the parabola opens. Positive \(p\) opens upwards (for vertical parabolas) or rightwards (for horizontal parabolas), while negative \(p\) opens downwards or leftwards.

Learn how to determine the equation of a parabola given its focus at (4, 2) and directrix y = -4. Understand the properties of parabolas, including the standard form and how to calculate key parameters. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation