https://file.aiquickdraw.com/mathbot/file/172532864783331rp7bji.jpg

The equation provided in the image is \( 7x^2 = 35 - 5y^2 \). Let's analyze and rewrite this equation in the standard form to identify the relevant information:

1. **Rewrite the equation**:  
   \[
   7x^2 + 5y^2 = 35
   \]

2. **Divide through by 35** to get it in standard form:
   \[
   \frac{x^2}{5} + \frac{y^2}{7} = 1
   \]

This is the equation of an ellipse in standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where:
- \( a^2 = 5 \) and \( b^2 = 7 \).

Given that \( b^2 > a^2 \), the ellipse is vertically oriented.

### Now, let's fill in the blanks:

1. **Center**: The center of the ellipse is at the origin, so the center is \( (0, 0) \).

2. **Vertices (Vert)**:  
   - For the vertically oriented ellipse, the vertices are located at \( (0, \pm b) \).  
   - Since \( b = \sqrt{7} \), the vertices are \( (0, \pm \sqrt{7}) \).

3. **Co-Vertices (CV)**:  
   - The co-vertices are located at \( (\pm a, 0) \).  
   - Since \( a = \sqrt{5} \), the co-vertices are \( (\pm \sqrt{5}, 0) \).

4. **Foci (Foci)**:  
   - The distance from the center to each focus \( c \) is given by \( c = \sqrt{b^2 - a^2} \).  
   - Here, \( c = \sqrt{7 - 5} = \sqrt{2} \).  
   - The foci are located at \( (0, \pm \sqrt{2}) \).

### Final Answers:
- **Center**: \( (0, 0) \)
- **Vertices**: \( (0, \pm \sqrt{7}) \)
- **Co-Vertices**: \( (\pm \sqrt{5}, 0) \)
- **Foci**: \( (0, \pm \sqrt{2}) \)

Would you like more details on any part of this solution? 

Here are 5 related questions to expand your understanding:
1. How do you derive the formula for the distance to the foci \( c \) for any ellipse?
2. What is the significance of the orientation of an ellipse in relation to its major and minor axes?
3. How would the equation change if the ellipse were horizontally oriented instead of vertically?
4. Can you derive the equation of the ellipse if the center is shifted to a point other than the origin?
5. How would you find the eccentricity of this ellipse, and what does it represent?

**Tip**: Always start by putting any conic section equation into its standard form to easily identify its key characteristics such as the center, vertices, and foci.

Understand the equation 7x^2 = 35 - 5y^2 and its transformation into the standard form of an ellipse. Learn about the ellipse's center, vertices, co-vertices, and foci. Suitable for advanced high school students and those interested in conic sections.

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