# 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

**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.**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`

**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)`

**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.