Question

Let \( P \) be the point on the surface \[ z = \sqrt{x^2 + y^2} \] closest to the point \( (4,2,0) \). Then the square of the distance between the origin and \( P \) is ................

Hint:

To find the point on a surface closest to another point, minimize the square of the distance function by using partial derivatives and solving for the critical points.

Answer 1

Correct Answer: 10.1 - 9.9

Step 1: Understanding the problem.
The surface is defined by the equation \( z = \sqrt{x^2 + y^2} \), and we are looking for the point \( P \) on this surface that is closest to the point \( (4, 2, 0) \). We are also tasked with finding the square of the distance between the origin and \( P \). The distance between two points in 3D space is given by: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Step 2: Setting up the objective function.
Let \( P = (x, y, z) \) be the point on the surface closest to \( (4, 2, 0) \). The distance between \( P \) and \( (4, 2, 0) \) is given by: \[ D(x, y) = \sqrt{(x - 4)^2 + (y - 2)^2 + (z)^2} \] Since \( z = \sqrt{x^2 + y^2} \), we substitute this into the equation for the distance: \[ D(x, y) = \sqrt{(x - 4)^2 + (y - 2)^2 + (x^2 + y^2)} \]
Step 3: Minimizing the distance function.
To find the point \( P \) that minimizes the distance, we minimize \( D^2(x, y) \), which eliminates the square root and simplifies the calculation: \[ D^2(x, y) = (x - 4)^2 + (y - 2)^2 + (x^2 + y^2) \] Now, we expand the terms: \[ D^2(x, y) = (x^2 - 8x + 16) + (y^2 - 4y + 4) + (x^2 + y^2) \] \[ D^2(x, y) = 2x^2 + 2y^2 - 8x - 4y + 20 \]
Step 4: Taking partial derivatives.
We take the partial derivatives of \( D^2(x, y) \) with respect to \( x \) and \( y \), and set them to zero to find the critical points: \[ \frac{\partial D^2}{\partial x} = 4x - 8 = 0 \quad \Rightarrow \quad x = 2 \] \[ \frac{\partial D^2}{\partial y} = 4y - 4 = 0 \quad \Rightarrow \quad y = 1 \]
Step 5: Substituting into the surface equation.
Now that we have \( x = 2 \) and \( y = 1 \), we substitute these values into the equation \( z = \sqrt{x^2 + y^2} \): \[ z = \sqrt{2^2 + 1^2} = \sqrt{5} \]
Step 6: Calculating the square of the distance.
Finally, we calculate the square of the distance between the origin \( (0, 0, 0) \) and \( P(2, 1, \sqrt{5}) \): \[ \text{Distance}^2 = 2^2 + 1^2 + (\sqrt{5})^2 = 4 + 1 + 5 = 10 \] Step 7: Conclusion.
Therefore, the square of the distance between the origin and \( P \) is 10.