https://file.aiquickdraw.com/mathbot/file/17261537909758phnila1.jpg

Let's go over the math questions presented in the image:

### Part A: Math Concept Check - Fill in the blanks
1. **An ellipse is the set of all points in the plane for which the sum of the distances from two fixed points F and F' is constant. The points F and F' are called the _foci_ of the ellipse.**
   
2. **The graph of the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a > b > 0\) is an ellipse with vertices _(\(±a, 0\))_ and foci _(\(±c, 0\))_, where \(c = \sqrt{a^2 - b^2}\).**
   
3. **The graph of \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) is an ellipse with _(\(0, ±a\))_ and foci _(\(0, ±c\))_, where \(c = \sqrt{a^2 - b^2}\).**
   
4. **Sketch the graph of the ellipse on #3 and label it.**
   
5. **Sketch the graph of the ellipse on #2 and label it.**

### Part B: Find the key characteristics for given ellipses
6. **For the equation \(2x^2 + 4y^2 = 16\), the center is at the origin \((0,0)\), the lengths of the major and minor axes can be found from the equation's structure.**
   
   - Simplify the equation:
   \[
   \frac{x^2}{8} + \frac{y^2}{4} = 1
   \]
   Major axis = 2\(\sqrt{8}\), minor axis = 2\(\sqrt{4}\).
   
7. **For the equation \(16x^2 + 25y^2 = 1600\), simplify and solve as in problem 6.**

8. **For the equation \(x^2 + 25y^2 = 21\), simplify and solve.**

### Part C: Determine if the equation represents an ellipse, point, or empty set
9. **For \(4x^2 + y^2 - 8x + 2y + 5 = 0\)**, simplify the equation to see what it represents.

10. **For \(2x^2 + 3y^2 + 8x - 6y + 20 = 0\), analyze similarly.**

### Part D: Ellipse Properties - Plot the ellipses with the given properties
11-15. **Plot and analyze the ellipses with the specific properties.**
 
Do you want more details on how to solve one of these specific problems or do you need further explanations?

Here are 5 related questions to expand your understanding:
1. How do you derive the formula for the foci of an ellipse?
2. What are the steps for converting a general conic equation into standard form for an ellipse?
3. How do you distinguish between a circle and an ellipse based on their equations?
4. How can you determine the orientation (horizontal vs. vertical) of an ellipse from its equation?
5. What is the significance of the eccentricity in an ellipse?

**Tip:** Always simplify the given equation of a conic to its standard form to identify key characteristics like vertices, foci, and axes lengths.

An ellipse is the set of all points in the plane for which the sum of the distances from two fixed points F and F' is constant. Find characteristics and graph specific ellipses, solve for key elements in ellipse equations, and determine if given conic equations represent an ellipse, point, or empty set.

This detailed guide covers how to find the vertices, foci, axes lengths, and more from ellipse equations. Learn to analyze conic sections, graph ellipses, and simplify complex ellipse equations. Perfect for high school students in Grades 10-12.

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