Consider the following statements about the surface

x squared equals 4 minus 4 y squared plus z squared

I. Axis of symmetry is the y-axis.

II. The surface is called hyperboloid of one sheet

III. The surface is called hyperboloid of two sheets

IV. Axis of symmetry is z-axis

Which of the following is true?

To analyze the given surface described by the equation \( x^2 = 4 - 4y^2 + z^2 \), we can rearrange it to a more familiar form. The equation can be rewritten as:

\[
\frac{x^2}{4} + \frac{4y^2 - z^2}{4} = 1
\]

This corresponds to the standard form of a hyperboloid, specifically:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1
\]

where \( a^2 = 4 \), \( b^2 = 1 \), and \( c^2 = 4 \). 

### Evaluating the Statements

I. **Axis of symmetry is the y-axis.**  
This is **false**. The axis of symmetry for a hyperboloid of one sheet like this one is along the x-axis and the z-axis.

II. **The surface is called hyperboloid of one sheet.**  
This is **true**. The form of the equation confirms it is a hyperboloid of one sheet.

III. **The surface is called hyperboloid of two sheets.**  
This is **false**. The given equation represents a hyperboloid of one sheet, not two sheets.

IV. **Axis of symmetry is the z-axis.**  
This is **true**. The hyperboloid has symmetry along the z-axis as well as the x-axis.

### Conclusion

Based on this analysis, the true statements are:

- II (The surface is called hyperboloid of one sheet)
- IV (Axis of symmetry is the z-axis)

### Summary

The correct answers are II and IV.

Would you like further details on any of these points? Here are some related questions you might consider:

1. What are the general characteristics of hyperboloids?
2. How do hyperboloids of one and two sheets differ?
3. Can you provide examples of real-world applications for hyperboloids?
4. What is the geometric significance of the axes of symmetry in these surfaces?
5. How can the standard forms of conic sections help in identifying different surfaces? 

**Tip:** When analyzing 3D surfaces, it's often useful to visualize them with graphing software to see their structure and axes of symmetry.

Consider the following statements about the surface x squared equals 4 minus 4 y squared plus z squared. I. Axis of symmetry is the y-axis. II. The surface is called hyperboloid of one sheet. III. The surface is called hyperboloid of two sheets. IV. Axis of symmetry is z-axis. Which of the following is true?

Explore the properties of the surface defined by x^2 = 4 - 4y^2 + z^2. Understand its classification as a hyperboloid of one sheet, the true statements regarding its axis of symmetry, and the key formulas involved. Ideal for high school students.

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