To find the points on the surface defined by the equation y² = 4xz that are closest to the origin (0, 0, 0), we need to minimize the distance from a point (x, y, z) on the surface to the origin.
The distance D from the origin to a point (x, y, z) is given by:
D = √(x² + y² + z²)
However, for minimization purposes, we can minimize the square of the distance to avoid dealing with the square root:
D² = x² + y² + z²
Now substituting y² = 4xz into the distance formula allows us to express everything in terms of x and z:
D² = x² + 4xz + z²
This is a function of two variables, D²(x, z). To find the minimum, we take the partial derivatives with respect to x and z and set them to zero.
First, we calculate the partial derivative with respect to x:
∂D²/∂x = 2x + 4z
Setting it to zero gives:
2x + 4z = 0 → x = -2z
Now, we calculate the partial derivative with respect to z:
∂D²/∂z = 4x + 2z
Setting this equal to zero gives:
4x + 2z = 0 → z = -2x
Now, we have two equations:
- x = -2z
- z = -2x
Substituting z = -2x into the first equation:
x = -2(-2x) → x = 4x
This leads us to realize that x must be zero:
4x = 0 → x = 0
Using x = 0 in one of our earlier relationships, we substitute back:
z = -2(0) = 0
y² = 4(0)z = 0 → y = 0
Thus, the only point on the surface y² = 4xz that is closest to the origin is:
(0, 0, 0)
To confirm, we can check that this point satisfies the original equation:
0² = 4(0)(0) → 0 = 0
Therefore, the point that is closest to the origin is indeed (0, 0, 0).