
Find and identify the traces of the quadric surface 
x2 + y2 − z2 = 49
 given the plane.
x = k
Find the trace.

We are given the quadric surface \(x^2 + y^2 - z^2 = 49\) and asked to find and identify the trace in the plane \(x = k\).

### Step 1: Substitute \(x = k\) into the equation of the surface
The general equation of the quadric surface is:
\[
x^2 + y^2 - z^2 = 49
\]
To find the trace on the plane \(x = k\), substitute \(x = k\) into this equation:
\[
k^2 + y^2 - z^2 = 49
\]

### Step 2: Rearrange the equation
Rearranging this equation gives:
\[
y^2 - z^2 = 49 - k^2
\]
This is the equation of a **hyperbola** in the \(yz\)-plane, provided that \(49 - k^2 > 0\). This will happen when \( |k| < 7 \) because \(k^2\) must be less than 49 for the right-hand side to be positive.

### Step 3: Analyze different cases for \(k\)
1. **When \( |k| < 7 \)**: 
   The equation \(y^2 - z^2 = 49 - k^2\) represents a **hyperbola** in the \(yz\)-plane.
   
2. **When \( |k| = 7 \)**:
   The equation becomes \(y^2 - z^2 = 0\), or equivalently \(y^2 = z^2\). This represents two intersecting lines: \(y = z\) and \(y = -z\).

3. **When \( |k| > 7 \)**:
   The right-hand side \(49 - k^2\) becomes negative, so the equation becomes \(y^2 - z^2 = \text{negative number}\). This has no real solutions, meaning there is no trace in this case.

### Conclusion:
- For \( |k| < 7 \), the trace is a **hyperbola**.
- For \( |k| = 7 \), the trace consists of two **intersecting lines**.
- For \( |k| > 7 \), there is **no real trace**.

Would you like further details on any of these cases or any clarifications?

Here are 5 related questions:
1. What is the general form of a hyperbola in the \(yz\)-plane?
2. How do you identify whether a trace is a hyperbola or an ellipse?
3. Can a trace of a quadric surface be an ellipse under certain conditions?
4. How does changing the constant value on the right-hand side affect the trace?
5. What is the significance of the lines formed when \( |k| = 7 \)?

**Tip**: For conic sections, knowing how to rewrite and manipulate the equation helps you identify the type of curve formed in any plane section.

Find and identify the traces of the quadric surface x^2 + y^2 - z^2 = 49 given the plane x = k. Find the trace.

Learn how to identify the trace of the quadric surface x^2 + y^2 − z^2 = 49 in the plane x = k. This involves analyzing the conic section formed, depending on the value of k, which results in a hyperbola, intersecting lines, or no real trace. Suitable for advanced mathematics students at the college level.

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