How do you find the points on the surface defined by the equation y² = 4xz that are closest to the origin?

To find the points on the surface defined by the equation y² = 4xz that are closest to the origin (0, 0, 0), we can follow these steps:

1. Understand the Equation: The given equation y² = 4xz represents a parabolic cylinder in a three-dimensional coordinate system.
2. Distance to the Origin: The distance D from any point (x, y, z) on this surface to the origin is given by the formula: D = sqrt(x² + y² + z²). To minimize the distance, it is often easier to minimize the square of the distance: D² = x² + y² + z².
3. Substitute the Surface Equation: We need to express D² in terms of two variables. Using the constraint y² = 4xz, we replace y in the distance formula. Since y = ±2√(xz), we get:
• D² = x² + (±2√(xz))² + z²
• D² = x² + 4xz + z²
4. Setting Up the Function To Minimize: Now we need to minimize the function: f(x, z) = x² + 4xz + z².
5. Calculate Partial Derivatives: To find the minimum, we take the partial derivatives of f(x, z) with respect to x and z and set them to zero:
• ∂f/∂x = 2x + 4z = 0
• ∂f/∂z = 4x + 2z = 0
6. Solving the System of Equations: From the first equation, we can express z in terms of x:
• 4z = -2x ⇒ z = -0.5x
7. Substituting this expression for z into the second equation:
• 4x + 2(-0.5x) = 0 ⇒ 4x - x = 0 ⇒ 3x = 0 ⇒ x = 0
8. Now, substituting x = 0 back into the expression for z:
• z = -0.5(0) = 0
9. Finally, substituting x = 0 into the equation for y gives:
• y² = 4(0)(0) = 0 ⇒ y = 0
10. Conclusion: The closest point on the surface y² = 4xz to the origin is (0, 0, 0). This means that the point closest to the origin on this parabolic cylinder is:
• (0, 0, 0).

To summarize, the point on the surface y² = 4xz that is closest to the origin is (0, 0, 0).