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The given hyperbolic paraboloid equation is:

\[
z = y^2 - x^2
\]

Let's go through each part of the problem one by one.

### Part (a) 
**Find the equation of the hyperbolic trace in the plane \( z = 4 \).**

To find the trace in the plane \( z = 4 \), we substitute \( z = 4 \) into the equation of the hyperbolic paraboloid:

\[
4 = y^2 - x^2
\]

Rearranging the equation:

\[
y^2 - x^2 = 4
\]

This is the equation of a hyperbola in the \( xy \)-plane.

---

### Part (b) 
**Find the vertices of the hyperbola in part (a).**

The standard form of a hyperbola is:

\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\]

By comparing this with \( y^2 - x^2 = 4 \), we rewrite it as:

\[
\frac{y^2}{4} - \frac{x^2}{4} = 1
\]

Here, \( a^2 = 4 \), so \( a = 2 \). The vertices of a hyperbola are located at \( (0, \pm a) \). Therefore, the vertices are:

\[
(0, 2) \quad \text{and} \quad (0, -2)
\]

---

### Part (c) 
**Find the foci of the hyperbola in part (a).**

For a hyperbola, the distance to the foci is given by:

\[
c = \sqrt{a^2 + b^2}
\]

Here, \( a^2 = 4 \) and \( b^2 = 4 \), so:

\[
c = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

Thus, the foci are located at \( (0, \pm c) \), or:

\[
(0, 2\sqrt{2}) \quad \text{and} \quad (0, -2\sqrt{2})
\]

---

### Part (d) 
**Describe the orientation of the focal axis of the hyperbola in part (a) relative to the coordinate axes.**

The hyperbola has the form \( \frac{y^2}{4} - \frac{x^2}{4} = 1 \), which is a vertically oriented hyperbola. The focal axis is along the \( y \)-axis because the \( y^2 \)-term comes first and has a positive coefficient.

Therefore, the focal axis is **parallel to the \( y \)-axis**.

---

Would you like to explore any further details on these steps?

Here are some related questions that might help deepen understanding:

1. What is the significance of the focal axis in determining the shape of the hyperbola?
2. How would the trace equation change if we chose a different plane for \( z \)?
3. Can the orientation of the hyperbola change if the \( z \)-value is negative?
4. How would you derive the asymptotes for the hyperbola from the given equation?
5. How does the shape of the hyperbolic paraboloid change as the \( z \)-value varies?

**Tip**: Always identify whether the hyperbola opens horizontally or vertically by looking at the variable with the positive term in the equation.

These exercises refer to the hyperbolic paraboloid z = y^2 - x^2.

43. (a) Find an equation of the hyperbolic trace in the plane z = 4.
(b) Find the vertices of the hyperbola in part (a).
(c) Find the foci of the hyperbola in part (a).
(d) Describe the orientation of the focal axis of the hyperbola in part (a) relative to the coordinate axes.

Solve problems related to the hyperbolic paraboloid z = y^2 - x^2. Learn how to find the trace in the z=4 plane, calculate the vertices and foci of the hyperbola, and understand the orientation of the focal axis. This exercise covers key concepts in conic sections and hyperbolas, ideal for advanced high school or undergraduate students.

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