The image contains two math problems. I will solve each in detail below.

Question 3: Find the hyperbolic paraboloid

The equation of a hyperbolic paraboloid generally takes the form:
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}$ or
$\frac{y^2}{b^2} - \frac{x^2}{a^2} = \frac{z}{c}.$

From the given options, the correct answer corresponds to: $y^2 + 1 = \frac{x^2}{4} - \frac{z^2}{9}.$ This equation is rearranged to clearly express the hyperbolic paraboloid structure.

Question 4: Equation of the line

We are tasked to find the equation of a line passing through $(7, 6, -3)$ and parallel to the given line:
$x = 5 - 8t, \; y = 7 + 3t, \; z = -12t.$

A line parallel to another has the same direction vector. From the parametric equations of the given line, the direction vector is: $\vec{d} = \langle -8, 3, -12 \rangle.$

Using the point $(7, 6, -3)$, the parametric equation of the required line is: $x = 7 - 8t, \; y = 6 + 3t, \; z = -3 - 12t.$

Thus, the correct option is: $x = 7 - 8t, \; y = 6 + 3t, \; z = -3 - 12t.$

Let me know if you'd like further details or clarifications.

Related Questions:

1. What is the difference between an elliptic paraboloid and a hyperbolic paraboloid?
2. How can you determine the equation of a line given two points?
3. What are the key features of a hyperbolic paraboloid in 3D geometry?
4. How do you derive a line parallel to another in vector form?
5. Can hyperbolic paraboloids occur naturally? If so, where?

Tip: When solving equations involving 3D shapes or lines, visualizing the geometry helps clarify the relationships.