Question

Find the point on the sphere x squaredplusleft parenthesis y minus 4 right parenthesis squaredplusleft parenthesis z minus 8 right parenthesis squaredequals9 nearest to

a. the​ xy-plane.
b. the point left parenthesis 0 comma 8 comma 8 right parenthesis.

Answer 1

To find the point on the sphere nearest to the xy-plane, substitute z = 0 into the equation. To find the point on the sphere nearest to a given point, minimize the distance formula using the equation of the sphere.

Step-by-step explanation:

To find the point on the sphere nearest to the xy-plane, we need to find the point on the sphere with the smallest z-coordinate. Since the equation of the sphere is given as x^2 + (y - 4)^2 + (z - 8)^2 = 9, we can substitute z = 0 into the equation and solve for x and y. Plugging z = 0 into the equation, we get x^2 + (y - 4)^2 + (0 - 8)^2 = 9, which simplifies to x^2 + (y - 4)^2 = 2. Solving this equation gives us two possible points: (2, 4) and (-2, 4). The point nearest to the xy-plane is (-2, 4, 0).

To find the point on the sphere nearest to the point (0, 8, 8), we need to find the point on the sphere with the smallest distance to this point. We can use the distance formula to find the distance between the point (x, y, z) on the sphere and the point (0, 8, 8). The distance formula is given by D = sqrt((x - 0)^2 + (y - 8)^2 + (z - 8)^2). We want to minimize this distance, so we can minimize D^2. Substituting the equation of the sphere into D^2, we get D^2 = (x^2 + (y - 4)^2 + (z - 8)^2) - ((0 - 0)^2 + (8 - 8)^2 + (8 - 8)^2). Simplifying this equation gives us D^2 = 9 - 0 = 9. Therefore, the point nearest to (0, 8, 8) on the sphere is any point on the sphere that satisfies x^2 + (y - 4)^2 + (z - 8)^2 = 9.

Answer 2

The center of the sphere of radius 3 is (0, 4, 8).

a) The point nearest the plane z=0, will be (0, 4, 5), one radius length below the center toward the xy-plane.

b) The line between the sphere's center and the given point is (x, z) = (0, 8). As in part (a), the closest point on that line to the given point is one radius-length from center along that line, or (0, 7, 8).