Question

The end points of a diameter of a sphere are P(1, 2, 3) and Q(5, 0, -1). Then the coordinates of the point on the sphere farthest from the origin is

• A. $(2 + \frac{8}{\sqrt{17}}, 3 + \frac{2}{\sqrt{17}}, 3 + \frac{2}{\sqrt{17}})$
• B. $(9 + \frac{3}{\sqrt{17}}, 5 + \frac{2}{\sqrt{17}}, 7 + \frac{2}{\sqrt{17}})$
• C. $(5 + \frac{2}{\sqrt{11}}, 5 + \frac{2}{\sqrt{11}}, 3 + \frac{2}{\sqrt{11}})$
• D. $(3 + \frac{9}{\sqrt{11}}, 1 + \frac{3}{\sqrt{11}}, 1 + \frac{3}{\sqrt{11}})$

Answer 1

Answer: (D)

$(3 + \frac{9}{\sqrt{11}}, 1 + \frac{3}{\sqrt{11}}, 1 + \frac{3}{\sqrt{11}})$

Answer 2

1. Find the center and radius of the sphere: The center of the sphere C is the midpoint of the diameter PQ. $C = \left(\frac{1+5}{2}, \frac{2+0}{2}, \frac{3+(-1)}{2}\right) = \left(\frac{6}{2}, \frac{2}{2}, \frac{2}{2}\right) = (3, 1, 1)$.

The radius $r$ is half the length of the diameter PQ. Diameter length $d = \sqrt{(5-1)^2 + (0-2)^2 + (-1-3)^2} = \sqrt{4^2 + (-2)^2 + (-4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6$. So, the radius is $r = \frac{6}{2} = 3$.

2. Find the point on the sphere farthest from the origin: The point on the sphere farthest from the origin lies on the line passing through the origin O(0,0,0) and the center of the sphere C(3,1,1). This point is located at a distance $r$ from the center C, on the side away from the origin.

The vector from the origin to the center is $\vec{OC} = (3-0, 1-0, 1-0) = (3, 1, 1)$. The magnitude of this vector is $|\vec{OC}| = \sqrt{3^2 + 1^2 + 1^2} = \sqrt{9 + 1 + 1} = \sqrt{11}$.

The unit vector in the direction of $\vec{OC}$ is $\hat{u}_{OC} = \frac{\vec{OC}}{|\vec{OC}|} = \left(\frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}}\right)$.

The coordinates of the point X on the sphere farthest from the origin are given by: $\vec{OX} = \vec{OC} + r \cdot \hat{u}_{OC}$ $\vec{OX} = (3, 1, 1) + 3 \cdot \left(\frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}}\right)$ $\vec{OX} = (3, 1, 1) + \left(\frac{9}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{3}{\sqrt{11}}\right)$ $\vec{OX} = \left(3 + \frac{9}{\sqrt{11}}, 1 + \frac{3}{\sqrt{11}}, 1 + \frac{3}{\sqrt{11}}\right)$.