https://file.aiquickdraw.com/mathbot/file/1727644723904d4g22riw.jpg

The equation given is:

\[
\frac{(x+5)^2}{9} + \frac{(y-3)^2}{16} = 1
\]

This is the standard form of an ellipse equation, where \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), with center \((h, k)\), and semi-major axis \(a\) and semi-minor axis \(b\). 

For the given equation:
- The center is \((-5, 3)\).
- The value under the \(x\)-term is 9, so \(a^2 = 9\) which gives \(a = 3\).
- The value under the \(y\)-term is 16, so \(b^2 = 16\) which gives \(b = 4\).

Since \(b > a\), the ellipse is vertically oriented.

Thus, the correct answer is **Ellipse**.

Would you like further details on how this conclusion was reached or explanations on specific terms?

Here are 5 related questions to consider:
1. How would the equation change if it described a circle?
2. What is the difference between a parabola and an ellipse in terms of their equations?
3. How can you determine the orientation of an ellipse from its equation?
4. What are the foci of an ellipse, and how would you calculate them for this equation?
5. How would the equation change if the ellipse was centered at the origin?

**Tip**: The denominator under the squared terms in ellipse equations indicates the square of the semi-axes' lengths.

Which shape is defined by the equation (x+5)^2/9 + (y-3)^2/16 = 1?

Determine the shape defined by the equation (x+5)^2/9 + (y-3)^2/16 = 1. Learn how to identify conic sections and solve ellipse equations with step-by-step solutions. This problem covers the key mathematical concept of conic sections and is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation