Use traces to sketch the surface.
9x2 − y2 + z2 = 0
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Identify the surface.
elliptic paraboloid
elliptic cone    
elliptic cylinder
ellipsoid
parabolic cylinder
hyperbolic paraboloid
hyperboloid of two sheets
hyperboloid of one sheet

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**.

---

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.

Use traces to sketch the surface.
9x^2 − y^2 + z^2 = 0

Identify the surface.
elliptic paraboloid
elliptic cone
elliptic cylinder
ellipsoid
parabolic cylinder
hyperbolic paraboloid
hyperboloid of two sheets
hyperboloid of one sheet

Learn how to sketch and identify the surface described by the quadratic equation 9x^2 − y^2 + z^2 = 0 using traces in various planes. This problem introduces the concept of quadratic surfaces, focusing on the elliptic cone. Suitable for college-level students studying analytic geometry or multivariable calculus.

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