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We are tasked with finding the point(s) on the surface \( z^2 = xy + 1 \) that are closest to the point \( (10, 8, 0) \).

### Problem breakdown:
1. The surface is given by the equation \( z^2 = xy + 1 \), which describes a relation between the coordinates \( (x, y, z) \).
2. We are trying to minimize the distance between any point \( (x, y, z) \) on this surface and the point \( (10, 8, 0) \). 

### Step 1: Distance Formula
The distance \( D \) between a point \( (x, y, z) \) on the surface and \( (10, 8, 0) \) is given by the Euclidean distance formula:
\[
D = \sqrt{(x - 10)^2 + (y - 8)^2 + z^2}
\]
Since minimizing \( D \) is equivalent to minimizing \( D^2 \), we can work with:
\[
D^2 = (x - 10)^2 + (y - 8)^2 + z^2
\]
But from the surface equation \( z^2 = xy + 1 \), we can substitute \( z^2 \) as \( xy + 1 \). So the expression for \( D^2 \) becomes:
\[
D^2 = (x - 10)^2 + (y - 8)^2 + (xy + 1)
\]
Thus, we now have the objective function:
\[
f(x, y) = (x - 10)^2 + (y - 8)^2 + xy + 1
\]
which we aim to minimize.

### Step 2: Gradient and Critical Points
To minimize the function \( f(x, y) \), we need to find the partial derivatives with respect to \( x \) and \( y \), set them to zero, and solve for \( x \) and \( y \).

Let me compute that.The critical point that minimizes the distance function is \( (x, y) = (8, 4) \).

### Step 3: Find the corresponding \( z \)-value
To find the \( z \)-coordinate corresponding to this point on the surface, we substitute \( x = 8 \) and \( y = 4 \) into the surface equation \( z^2 = xy + 1 \):
\[
z^2 = 8 \cdot 4 + 1 = 33
\]
Thus, \( z = \pm \sqrt{33} \).

### Final Answer:
The points on the surface that are closest to \( (10, 8, 0) \) are:
\[
(8, 4, \sqrt{33}) \quad \text{and} \quad (8, 4, -\sqrt{33})
\]

Do you want further details on any step, or any clarification?

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Here are 5 related questions:
1. How can we generalize this method for other surfaces?
2. What are the second-order conditions for confirming a local minimum?
3. What happens if the surface is defined by a different equation, such as a plane?
4. How does this method work in higher dimensions (e.g., in 4D space)?
5. Can the distance minimization problem be solved using Lagrange multipliers?

**Tip:** Minimizing the square of a distance often simplifies computations without affecting the result, as the square preserves the relative ordering of distances.

Find the point(s) on the surface z^2 = xy + 1 which are closest to the point (10, 8, 0).

Learn how to find the point(s) on the surface z^2 = xy + 1 that are closest to the point (10, 8, 0) using optimization techniques. This solution covers concepts like the Euclidean distance formula, partial derivatives, and surface equation minimization.

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