To find the points on the surface defined by the equation y² = 49xz that are closest to the origin, we need to minimize the distance from a point (x, y, z) on this surface to the origin (0, 0, 0).
The distance D from the origin to a point (x, y, z) can be expressed as:
D = sqrt(x² + y² + z²)
However, it is often easier to minimize the square of the distance D² = x² + y² + z² instead, as this avoids dealing with the square root. Thus, we aim to minimize:
D² = x² + y² + z²
Substituting the surface equation y² = 49xz into our distance formula gives us:
D² = x² + 49xz + z²
To find the minimum, we can set up a Lagrange multiplier or take partial derivatives, but a simpler approach is to express z in terms of x and y using the constraint:
z = rac{y²}{49x}
Substituting into our distance formula:
D² = x² + y² + rac{y^4}{49x²}
Next, we can differentiate this with respect to x and y, set the derivatives to zero, and solve the resulting equations for critical points. This will help us identify the points that minimize D².
After computing the derivatives and solving, we find that the points that are closest to the origin occur when x = 0, which leads us to the potential minima on the surface, giving us points such as (0, 0, 0) as valid candidates, or other symmetrical points based on the nature of the surface.
In summary, the points on the surface y² = 49xz that are closest to the origin are determined by evaluating the distance function while adhering to the given constraint. This involves using calculus to find critical points.