graph The intersection of the surface 𝑧2=𝑥24+𝑦2 and the plane 𝑧=0 resembles which of the following?

To find the intersection of the surface \( z^2 = x^4 + y^2 \) and the plane \( z = 0 \), we need to substitute \( z = 0 \) into the equation of the surface.

Starting with the equation of the surface:
\[
z^2 = x^4 + y^2
\]
Substitute \( z = 0 \) into the equation:
\[
0^2 = x^4 + y^2
\]
This simplifies to:
\[
x^4 + y^2 = 0
\]
Now, since \( x^4 \) and \( y^2 \) are both non-negative for all real values of \( x \) and \( y \), the only way the sum can equal zero is if both \( x^4 = 0 \) and \( y^2 = 0 \), which occurs when:
\[
x = 0 \quad \text{and} \quad y = 0
\]

Thus, the only point of intersection is at the origin \( (0, 0, 0) \).

### Conclusion:
The intersection of the surface \( z^2 = x^4 + y^2 \) and the plane \( z = 0 \) is a single point at the origin. Therefore, the intersection resembles a **point**.

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Would you like any further details or clarifications?

Here are some related questions to explore further:
1. How does the shape of the surface \( z^2 = x^4 + y^2 \) change for different values of \( z \)?
2. What would the intersection look like if the plane had a different value, such as \( z = 1 \)?
3. How would the equation change if the surface were \( z^2 = x^2 + y^2 \) instead of \( z^2 = x^4 + y^2 \)?
4. Can you find the critical points of the surface \( z^2 = x^4 + y^2 \)?
5. What is the significance of the higher-order term \( x^4 \) in the surface equation?

**Tip:** When analyzing surfaces and their intersections with planes, always check if substitution simplifies the problem to a basic geometric figure like a point, line, or curve.

The intersection of the surface z^2 = x^4 + y^2 and the plane z = 0 resembles which of the following?

Discover how to find the intersection of the surface z^2 = x^4 + y^2 and the plane z = 0. Learn the key algebraic and geometric concepts involved, including the behavior of higher-order terms and non-negative squares, with a step-by-step explanation. Suitable for college-level students studying calculus and geometry.

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