Finding the Closest Points on a Cone to a Given Point

In this article, we will explore the problem of finding the points on the cone z^2 = x^2 + y^2 that are closest to the point (1, 2, 0). This problem requires a good understanding of 3D geometry and the properties of cones.

Understanding the Cone Equation

The equation z^2 = x^2 + y^2 represents a cone in the 3D Cartesian coordinate system. This cone has its vertex at the origin (0, 0, 0) and extends infinitely in both the positive and negative directions along the z-axis.

The cross-section of the cone at any given z-value is a circle with a radius equal to the absolute value of z. In other words, the points on the cone satisfy the equation:

(x, y, z) | z^2 = x^2 + y^2

Finding the Closest Points

To find the points on the cone that are closest to the point (1, 2, 0), we need to minimize the distance between the cone and the given point. The distance between a point (x, y, z) on the cone and the point (1, 2, 0) can be calculated using the Euclidean distance formula:

d = sqrt((x - 1)^2 + (y - 2)^2 + z^2)

To find the minimum distance, we need to solve the following optimization problem:

Minimize d = sqrt((x - 1)^2 + (y - 2)^2 + z^2) Subject to z^2 = x^2 + y^2

This is a constrained optimization problem, and we can solve it using the method of Lagrange multipliers.

Solving the Problem Using Lagrange Multipliers

1. Set up the Lagrangian function: The Lagrangian function is defined as:

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

   where λ is the Lagrange multiplier.

2. Find the critical points: To find the critical points, we need to take the partial derivatives of the Lagrangian function with respect to x, y, z, and λ, and set them equal to zero:

   ∂L/∂x = 2(x - 1) - 2λx = 0 ∂L/∂y = 2(y - 2) - 2λy = 0 ∂L/∂z = 2z + 2λz = 0 ∂L/∂λ = z^2 - x^2 - y^2 = 0

3. Solve the system of equations: Solving the system of equations, we get the following critical points:

   (x, y, z) = (1, 2, 0) (x, y, z) = (0, 0, 0)

4. Evaluate the distance: Substituting the critical points into the distance formula, we get:

   For (x, y, z) = (1, 2, 0), the distance is d = sqrt((1 - 1)^2 + (2 - 2)^2 + 0^2) = 0. For (x, y, z) = (0, 0, 0), the distance is d = sqrt((0 - 1)^2 + (0 - 2)^2 + 0^2) = sqrt(5).

Therefore, the points on the cone z^2 = x^2 + y^2 that are closest to the point (1, 2, 0) are (1, 2, 0) and (0, 0, 0). The distance to both of these points is 0, as the point (1, 2, 0) lies on the cone.