The length of the latus rectum is 12 and the parabola opens to the left

To find the equation of a parabola with a given latus rectum length and direction, we use the standard form of the parabola that opens to the left. The general equation for a parabola opening to the left is:

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

Where:
- \((h, k)\) is the vertex of the parabola.
- \(p\) is the distance from the vertex to the focus (the focal length).
- The latus rectum length is given by \(4p\), which represents the length of the line segment perpendicular to the axis of symmetry, passing through the focus, and whose endpoints lie on the parabola.

### Given:
- The length of the latus rectum is 12, so:
\[
4p = 12 \implies p = \frac{12}{4} = 3
\]

Since the parabola opens to the left, \(p\) is negative. Thus, \(p = -3\).

### Equation of the Parabola:
Substituting \(p = -3\) into the general equation, we get:

\[
(y - k)^2 = -4(-3)(x - h)
\]
\[
(y - k)^2 = -12(x - h)
\]

This is the standard equation of the parabola with a latus rectum of 12, opening to the left.

Would you like to explore more details, or have any questions? Here are some related questions:

1. What is the focus of the parabola if the vertex is known?
2. How would the equation change if the parabola opened to the right?
3. Can you find the directrix of this parabola?
4. How does the value of \(p\) affect the shape of the parabola?
5. How can you find the endpoints of the latus rectum from the equation?

**Tip**: The sign of \(p\) determines the direction of the parabola (positive for right/up, negative for left/down).

The length of the latus rectum is 12 and the parabola opens to the left.

Discover how to find the equation of a parabola with a latus rectum length of 12 that opens to the left. This problem explores conic sections, specifically parabolas, and involves understanding the relationship between latus rectum length, focal length, and the direction of the parabola. Suitable for high school students in 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