To find the points on the surface defined by the equation y² = 64xz that are closest to the origin, we can use the method of Lagrange multipliers or minimize the squared distance from the origin. The squared distance from a point (x, y, z) to the origin is given by D = x² + y² + z².
We can substitute the surface equation into the distance formula. From the equation y² = 64xz, we can express z in terms of x and y as z = y² / (64x). By substituting this back into the distance formula, we get:
D = x² + y² + (y² / (64x))².
This leads us to minimize D with respect to x and y. To do this, we can take partial derivatives and set them to zero to find critical points.
After simplifying and solving the resulting equations, we can determine the values of (x, y, z) that yield the closest points on the surface to the origin. By substituting back into the surface’s equation, we ultimately derive the points. The solutions might yield multiple points, typically symmetric in nature.
Finally, we can evaluate these points and confirm their distances to find which are indeed the closest to the origin.