To identify the surface and sketch it using traces, we begin by analyzing the given equation:

$9x^2 - y^2 + z^2 = 0$

Step 1: Rearrange the equation.

We rewrite the equation in a more familiar form by isolating $y^2$ on one side: $9x^2 + z^2 = y^2$ or equivalently: $y^2 = 9x^2 + z^2$

Step 2: Analyze the equation.

This equation represents a cone. Specifically, it is a double-napped cone because it relates $y^2$ to a positive quadratic expression in $x^2$ and $z^2$, which is characteristic of an elliptic cone.

Step 3: Cross-sections (traces).

Let's verify this by analyzing traces in various planes.

• Trace in the plane $x = 0$: $y^2 = z^2$ This simplifies to $y = \pm z$, which represents two straight lines, indicating that in the $yz$-plane, we have intersecting lines.

• Trace in the plane $z = 0$: $y^2 = 9x^2$ This simplifies to $y = \pm 3x$, indicating two straight lines in the $xy$-plane.

• Trace in the plane $y = k$ (a constant): $k^2 = 9x^2 + z^2$ This is the equation of an ellipse, which confirms that the cross-section at a fixed $y$ value is an ellipse.

Step 4: Conclusion.

The surface described by the equation is an elliptic cone, as it satisfies the characteristics of a cone with elliptical cross-sections.

Final Answer:

The surface is an elliptic cone.

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Would you like more details or a visualization? Here are some related questions:

1. What is the difference between an elliptic cone and a hyperboloid of one sheet?
2. How would the surface change if the signs in the equation were different?
3. How can we derive the general form of a cone from the equation?
4. Can the same equation describe a different surface under some conditions?
5. What role do traces play in identifying 3D surfaces?

Tip: For quadratic surfaces, always check cross-sections (traces) in different planes to help identify the surface shape.