Let's break down the problem step by step.

1. Reduce the equation to a standard form:

We start with the equation:

$25x^2 - y + 5z^2 = 0$

Rearrange it to solve for $y$:

$y = 25x^2 + 5z^2$

This is a quadratic equation in $x$ and $z$, with a clear relationship between $x^2$ and $z^2$.

2. Classify the surface:

The equation $y = 25x^2 + 5z^2$ represents a paraboloid because:

• The $x^2$ and $z^2$ terms are squared, and there is no linear or mixed term.
• The coefficients of $x^2$ and $z^2$ are positive, so the surface opens upwards.

More specifically, since both squared terms are positive, this is an elliptic paraboloid.

3. Write an equation for the cross section at $z = 0$ using $x$ and $y$:

When $z = 0$, the equation becomes:

$y = 25x^2$

This represents a parabola in the $xy$-plane.

4. Write an equation for the cross section at $y = -2$ using $x$ and $z$:

When $y = -2$, substitute $y = -2$ into the original equation:

$-2 = 25x^2 + 5z^2$

Rearrange to isolate the equation in terms of $x$ and $z$:

$25x^2 + 5z^2 = -2$

This equation does not describe a real surface because the left-hand side is always non-negative (both $x^2$ and $z^2$ are non-negative), while the right-hand side is negative. Hence, there are no real solutions for this case.

5. Summary:

• The equation represents an elliptic paraboloid.
• The cross section at $z = 0$ is $y = 25x^2$ (a parabola).
• The cross section at $y = -2$ does not exist because it results in a negative sum on the left-hand side, making it impossible for real values of $x$ and $z$.

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Would you like more details on any part of the solution, or is there anything else you'd like to discuss?

Here are some related questions:

1. How does the sign of the coefficients in a quadratic equation affect the classification of a surface?
2. What are the key differences between an elliptic paraboloid and a hyperbolic paraboloid?
3. How do we interpret cross sections of surfaces in different planes?
4. How can we graph a paraboloid or a hyperboloid based on its equation?
5. What are some physical applications of elliptic paraboloids?

Tip: To visualize parabolas and other conic sections, it's helpful to plot their cross sections in both the $xy$- and $xz$-planes!