1. C(3, -5); vertical major axis of length 12; minor axis of length 8


Given the information:

1. Center of the ellipse: \( C(3, -5) \)
2. Vertical major axis of length 12
3. Minor axis of length 8

We can determine the equation of the ellipse.

### Step 1: Identify the semi-major and semi-minor axes lengths.
- The major axis is vertical, so the length of the semi-major axis \( a \) is half of 12:
  \[
  a = \frac{12}{2} = 6
  \]
- The minor axis is horizontal, so the length of the semi-minor axis \( b \) is half of 8:
  \[
  b = \frac{8}{2} = 4
  \]

### Step 2: Write the equation of the ellipse.
Since the major axis is vertical and the center is at \( (h, k) = (3, -5) \), the general form of the ellipse's equation is:
\[
\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1
\]
Substituting the values \( h = 3 \), \( k = -5 \), \( a = 6 \), and \( b = 4 \), the equation becomes:
\[
\frac{(x - 3)^2}{4^2} + \frac{(y + 5)^2}{6^2} = 1
\]
Simplifying further:
\[
\frac{(x - 3)^2}{16} + \frac{(y + 5)^2}{36} = 1
\]

This is the equation of the ellipse.

Would you like to explore any more details or have any questions?

### Related Questions:
1. How do you determine the foci of an ellipse?
2. What is the eccentricity of this ellipse?
3. How does the orientation of the major axis affect the equation of the ellipse?
4. Can you find the coordinates of the vertices of this ellipse?
5. How would the equation change if the major axis were horizontal instead of vertical?

### Tip:
Remember, the semi-major axis is always associated with the larger denominator under the variable term in the ellipse equation.

C(3, -5); vertical major axis of length 12; minor axis of length 8

Explore how to derive the equation of an ellipse with a center at (3, -5), a vertical major axis of length 12, and a minor axis of length 8. This problem covers key concepts in coordinate geometry and the standard form of an ellipse, suitable for 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