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The given problem involves finding the equations for:

1. The tangent plane to the surface defined by \( y = x^2 - z^2 \) at the point \( (5, 21, 2) \).
2. The normal line to the surface at the same point.

Let's go step-by-step:

### Step 1: Find the Tangent Plane
The equation of the tangent plane to the surface at a point can be written as:
\[
f_x(x_0, z_0)(x - x_0) + f_y(x_0, z_0)(y - y_0) + f_z(x_0, z_0)(z - z_0) = 0
\]
where \( f_x \), \( f_y \), and \( f_z \) are the partial derivatives of the function, and \( (x_0, y_0, z_0) \) is the point on the surface.

The surface is defined by:
\[
f(x, y, z) = y - (x^2 - z^2) = 0
\]

The partial derivatives are:
\[
f_x = -2x, \quad f_y = 1, \quad f_z = 2z
\]

At the point \( (x_0, y_0, z_0) = (5, 21, 2) \):
\[
f_x(5, 2) = -2(5) = -10, \quad f_y(5, 21, 2) = 1, \quad f_z(5, 2) = 2(2) = 4
\]

Thus, the equation of the tangent plane is:
\[
-10(x - 5) + 1(y - 21) + 4(z - 2) = 0
\]
Simplifying:
\[
-10x + 50 + y - 21 + 4z - 8 = 0
\]
\[
-10x + y + 4z + 21 = 0
\]

### Step 2: Find the Normal Line
The normal line is defined by the parametric equations:
\[
x(t) = x_0 + f_x t, \quad y(t) = y_0 + f_y t, \quad z(t) = z_0 + f_z t
\]
Substituting the values of the partial derivatives and the point \( (5, 21, 2) \):
\[
x(t) = 5 - 10t, \quad y(t) = 21 + t, \quad z(t) = 2 + 4t
\]

So the parametric equation for the normal line is:
\[
(x(t), y(t), z(t)) = (5 - 10t, 21 + t, 2 + 4t)
\]

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Would you like further clarification or more details on any step?

Here are five related questions:
1. How do you find the normal vector to a surface at a point?
2. What is the general formula for the tangent plane of a surface?
3. Why do we use partial derivatives in tangent plane equations?
4. How would the tangent plane equation change if the surface had a more complex form?
5. How do you interpret the parametric equation of the normal line geometrically?

**Tip:** The normal vector is crucial when finding both the tangent plane and the normal line, as it directly influences the orientation of the plane and the direction of the line.

Find the equations for the tangent plane and the normal line to the surface y = x^2 - z^2 at the point (5, 21, 2).

Find the equations for the tangent plane and normal line to the surface y = x^2 - z^2 at the point (5, 21, 2). This solution uses partial derivatives and includes the parametric equation of the normal line.

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