https://file.aiquickdraw.com/mathbot/file/17267809282622s8o63h1.jpg

The image contains a set of problems related to ellipses in the form of equations. Based on what I can interpret from the image, the four problems appear to be asking for properties of ellipses. Here's a clearer outline:

1. **Problem 1** (not fully visible): The question likely asks for the **eccentricity** of an ellipse.
2. **Problem 2**: The equation is \(25x^2 + 16y^2 = 400\). This is the equation of an ellipse.
3. **Problem 3**: The equation is \(\frac{(x - 1)^2}{100} + \frac{(y + 3)^2}{25} = 1\), which is an ellipse in standard form.
4. **Problem 4**: The equation is \(9x^2 + 4y^2 - 54x - 8y + 49 = 0\), which is the equation of a conic section (most likely an ellipse after completing the square).

For all these problems, the main objective is likely to analyze these ellipses by:
- **Finding the eccentricity**.
- **Identifying important geometric properties** such as semi-major and semi-minor axes, centers, and foci.

Would you like me to solve the equations for their properties step by step? I can start with any specific problem you'd like.

Also, here are 5 relative questions for expanding the topic:
1. How do you find the standard form of an ellipse from a general quadratic equation?
2. What is the geometric significance of the semi-major and semi-minor axes in an ellipse?
3. How do we determine the foci of an ellipse?
4. Can you explain the relationship between eccentricity and the shape of an ellipse?
5. How do transformations (shifts in \(x\) or \(y\)) affect the center of an ellipse?

**Tip:** When analyzing ellipses, always ensure the equation is in its standard form before calculating properties like eccentricity or axes lengths.

Solve the following problems related to ellipses:
1. Determine the eccentricity of the ellipse.
2. Given the ellipse equation 25x^2 + 16y^2 = 400, find its properties.
3. Given the ellipse equation (x - 1)^2 / 100 + (y + 3)^2 / 25 = 1, find its properties.
4. Given the equation 9x^2 + 4y^2 - 54x - 8y + 49 = 0, find the ellipse properties after simplifying.

Learn how to solve ellipse equations like 25x^2 + 16y^2 = 400, including finding the eccentricity, semi-major and semi-minor axes, and more. Perfect for Grades 10-12.

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