Finding the Closest Point on the Surface z^2 = xy + 1 to the Point (7, 8, 0)

In this article, we will solve the problem of finding the point(s) on the surface z^2 = xy + 1 that are closest to the given point (7, 8, 0).

To solve this problem, we will use the concepts of calculus and optimization. The goal is to find the point(s) on the surface that minimize the distance to the given point.

Step 1: Understand the problem

The surface is defined by the equation z^2 = xy + 1, and we need to find the point(s) on this surface that are closest to the point (7, 8, 0).

Step 2: Set up the optimization problem

Let's define the distance function d(x, y, z) between the point (x, y, z) on the surface and the point (7, 8, 0):

d(x, y, z) = sqrt((x - 7)^2 + (y - 8)^2 + z^2)

We need to minimize this distance function, subject to the constraint that the point (x, y, z) lies on the surface z^2 = xy + 1.

Step 3: Use the method of Lagrange multipliers

To solve this optimization problem, we can use the method of Lagrange multipliers. We will introduce a Lagrange multiplier λ and define the Lagrangian function:

L(x, y, z, λ) = d(x, y, z) + λ(z^2 - xy - 1)

Now, we need to find the critical points of the Lagrangian function, where the partial derivatives with respect to x, y, z, and λ are all equal to 0.

∂L/∂x = -(x - 7) / d(x, y, z) - λy = 0 ∂L/∂y = -(y - 8) / d(x, y, z) - λx = 0 ∂L/∂z = z / d(x, y, z) + 2λz = 0 ∂L/∂λ = z^2 - xy - 1 = 0

Solving this system of equations, we can find the critical points (x, y, z) that satisfy the surface equation and minimize the distance to the point (7, 8, 0).

Step 4: Analyze the critical points

After solving the system of equations, we may find one or more critical points that satisfy the surface equation and minimize the distance to the given point. These critical points represent the point(s) on the surface that are closest to (7, 8, 0).

It's important to note that the critical points found may not be the only points on the surface that are closest to (7, 8, 0). Depending on the specific surface and the given point, there may be multiple points on the surface that are equally close to the given point.

Conclusion

In this article, we have provided a detailed step-by-step solution to the problem of finding the point(s) on the surface z^2 = xy + 1 that are closest to the point (7, 8, 0). By using the method of Lagrange multipliers, we were able to set up and solve the optimization problem to find the critical points that minimize the distance to the given point.

This problem demonstrates the application of calculus and optimization techniques in solving geometric problems involving surfaces and points in 3D space. Understanding and applying these concepts can be valuable in various fields, such as engineering, physics, and computer graphics.