To find the points on the cone defined by the equation z = √(x² + y²) that are closest to the point (4, 2, 0), we will use the method of Lagrange multipliers for optimization.
First, we express the distance squared from a point (x, y, z) on the cone to the point (4, 2, 0) as:
D(x, y, z) = (x – 4)² + (y – 2)² + z²
Since we are constrained by the equation of the cone, we substitute z in our distance function:
D(x, y) = (x – 4)² + (y – 2)² + (√(x² + y²))²
This simplifies to:
D(x, y) = (x – 4)² + (y – 2)² + x² + y²
Now, to minimize this function, we can take partial derivatives with respect to x and y, then set them to zero:
∂D/∂x = 2(x – 4) + 2x = 0
∂D/∂y = 2(y – 2) + 2y = 0
Solving these equations gives us:
3x – 8 = 0 → x = 8/3
3y – 4 = 0 → y = 4/3
Now we calculate z using the cone’s equation:
z = √((8/3)² + (4/3)²) = √(64/9 + 16/9) = √(80/9) = (4√5)/3
Thus, the points on the cone closest to (4, 2, 0) are:
(8/3, 4/3, (4√5)/3)
In conclusion, through optimization techniques, we found the point on the cone that minimizes the distance to the given external point.