The surface given is the hyperbolic paraboloid described by the equation:

$z = x^2 - y^2$

Task:

We need to determine whether there exists a line on this surface that is parallel to the $x$-axis. If such a line exists, we need to find it; otherwise, we will prove that it is impossible.

Step-by-Step Approach:

1. Line parallel to the $x$-axis: A line parallel to the $x$-axis can be written in the parametric form as: $x = t, \quad y = y_0, \quad z = z_0$ where $t$ is the parameter that varies along the $x$-axis, and $y_0$ and $z_0$ are constants that define the specific line.

2. Substitute into the surface equation: For a point $(t, y_0, z_0)$ to lie on the surface $z = x^2 - y^2$, it must satisfy the surface equation. That is, we substitute $x = t$, $y = y_0$, and $z = z_0$ into the equation $z = x^2 - y^2$:

   $z_0 = t^2 - y_0^2$

3. Analyze the equation: From the above equation, we see that the $z$-coordinate $z_0$ depends on $t^2$. This means that as $t$ changes (i.e., as we move along the $x$-axis), $z_0$ also changes (specifically, it follows a parabolic form).

   Therefore, it is impossible to have a line parallel to the $x$-axis on the surface because the $z$-coordinate must vary with $t^2$, and thus cannot remain constant. A line parallel to the $x$-axis would require a constant $z$, which is not possible on this surface.

Conclusion:

There does not exist a line that lies on the surface and is parallel to the $x$-axis, because the $z$-coordinate will always vary as a function of $t^2$, which contradicts the condition for a line parallel to the $x$-axis.

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Do you want more details on this explanation, or have any questions?

Here are 5 related questions:

1. Can a line parallel to the $y$-axis lie on the hyperbolic paraboloid surface?
2. How do we find lines of intersection between the surface and other planes?
3. Can we use vector calculus to analyze the tangent planes to the surface?
4. What are the properties of hyperbolic paraboloid surfaces in general?
5. How would the surface change if the equation were $z = y^2 - x^2$?

Tip: When analyzing surfaces, using parametric forms of lines can help identify potential constraints or contradictions.