To find the points on the surface defined by the equation y² + 49xz = 0 that are closest to the origin, we will use the method of Lagrange multipliers. This technique finds the extrema of a function subject to a constraint.
Firstly, we need to express our goal mathematically. The distance from the origin (0, 0, 0) to a point (x, y, z) is given by the distance formula:
D = √(x² + y² + z²)
However, since we’ll be minimizing D, it is simpler to minimize the square of the distance D² = x² + y² + z². Hence, we want to:
Minimize: f(x, y, z) = x² + y² + z²
Subject to: g(x, y, z) = y² + 49xz = 0
Next, we’ll set up the Lagrange function:
ℒ(x, y, z, λ) = f(x, y, z) + λg(x, y, z)
Now, we can derive the system of equations:
- ∂ℒ/∂x = 2x + λ(49z) = 0
- ∂ℒ/∂y = 2y + λ(2y) = 0
- ∂ℒ/∂z = 2z + λ(49x) = 0
- ∂ℒ/∂λ = g(x, y, z) = 0
From the second equation, we can solve for λ:
If y ≠ 0: 2y(1 + λ) = 0 implies λ = -1
If y = 0: substitute into the constraint 49xz = 0; so either x = 0 or z = 0.
Using various combinations of these results, we can find conditions:
- If x = 0, y = 0, then g(0, 0, z) = 0 gives z = 0. Point: (0, 0, 0).
- If y = 0 and z = 0, then you can pick any value of x, yielding the line along the x-axis.
- If x = 0 with y ≠ 0, we can set z = - rac{y²}{49} based on substitution from the constraint.
Optimizing these, we find that points (0, y, -y²/49) are closest when y = ±7 since they minimize D:
- Coordinates: (0, 7, -1)
- Coordinates: (0, -7, -1)
Thus, the points on the surface defined by y² + 49xz = 0 that are closest to the origin are (0, 7, -1) and (0, -7, -1).