The points (2, 3,6) and (6, 3, !2) both lie on the sphere x

2
+ y
2
+ z
2
= 49. What is the shortest
path between them if travel is restricted to the surface of the sphere?

To find the shortest path between the two points on the surface of the sphere, we need to determine the length of the arc of the great circle that connects them.

First, let's find the equation of the sphere using the given points.

The general equation of a sphere is (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2, where (a, b, c) is the center of the sphere and r is its radius.

From the given points, we can make two equations:

(2 - a)^2 + (3 - b)^2 + (6 - c)^2 = 49
(6 - a)^2 + (3 - b)^2 + (-12 - c)^2 = 49

Simplifying these equations, we get:
a^2 - 4a + b^2 - 6b + c^2 - 8c + 36 = 0 ---(1)
a^2 - 12a + b^2 - 6b + c^2 + 12c + 108 = 0 ---(2)

Now, we can solve these two equations simultaneously to find the values of a, b, and c.

By subtracting equation (1) from equation (2), we eliminate the variables a and b, and we get:
8a - 20c + 72 = 0

Simplifying further, we have:
4a - 10c + 36 = 0

Next, we can express a in terms of c by isolating it:
4a = 10c - 36
a = (10c - 36) / 4
a = (5c - 18) / 2

Substituting the value of a in either equation (1) or (2), we can solve for b in terms of c:
[(5c - 18) / 2]^2 - 4(5c - 18) + b^2 - 6b + c^2 - 8c + 36 = 0

Expanding and simplifying, we have:
25c^2 - 180c + 324 - 20c + 72 + b^2 - 6b + c^2 - 8c + 36 = 0

Combining like terms, we get:
26c^2 - 208c + 432 + b^2 - 6b + 2c^2 - 14c + 36 = 0

Grouping the terms, we have:
28c^2 - 222c + 468 + b^2 - 6b = 0

We now have a system of two equations:
4a - 10c + 36 = 0
28c^2 - 222c + 468 + b^2 - 6b = 0

Solving this system of equations will give us the values of a, b, and c.

Once we have the values of a, b, and c, we can use the distance formula to find the distance between the two points on the surface of the sphere.

Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]

Substituting the coordinates of the two points, we can calculate the shortest path between them on the surface of the sphere.