To find the points on the cone defined by the equation z2 = x2 + y2 that are closest to the point (2, 2, 0), we will employ the method of Lagrange multipliers.
First, we recognize that the distance D from any point on the cone (x, y, z) to the point (2, 2, 0) can be expressed as:
D = √{(x - 2)2 + (y - 2)2 + z2}
However, it’s easier to minimize the square of the distance to avoid dealing with the square root:
D2 = (x - 2)2 + (y - 2)2 + z2
With the constraint given by the cone’s equation, we need to consider:
g(x, y, z) = z2 - (x2 + y2) = 0
Now, we will set up the method of Lagrange multipliers. We need to solve the following system of equations derived from the gradients:
∇f = λ∇g
Where f is our objective function:
f(x, y, z) = (x - 2)2 + (y - 2)2 + z2
Calculating the gradients:
∇f = (2(x - 2), 2(y - 2), 2z)
∇g = (-2x, -2y, 2z)
Setting these equal gives us the following equations:
2(x - 2) = -λ(2x)
2(y - 2) = -λ(2y)
2z = λ(2z)
From the third equation, if z ≠ 0, we can simplify and find:
λ = 1
Inserting λ = 1 into the first two equations gives:
x - 2 = -x → 2x = 2 → x = 1
y - 2 = -y → 2y = 2 → y = 1
So one candidate for the closest point is:
(1, 1, z)
Now substituting x = 1 and y = 1 into the cone’s equation:
z2 = 12 + 12 → z2 = 2 → z = ±√2
Thus, we have the two points on the cone that are closest to (2, 2, 0):
- (1, 1, √2)
- (1, 1, -√2)
To confirm these points indeed yield the minimum distance, you could calculate the second derivatives or simply check the distance values, but generally, these findings indicate the correct approach to this optimization problem.