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.