Centre of the Sphere Passing Through Three Points

To find the center of the sphere passing through the given points, we can use the concept of the circumcenter. The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle. In this case, we can consider the triangle formed by the three given points and find its circumcenter, which will give us the center of the sphere.

Step 1: Finding the Midpoints of the Sides

First, we need to find the midpoints of the sides of the triangle formed by the given points. The midpoints can be found by averaging the coordinates of the endpoints of each side.

Let's label the given points as A(2, 1, -1), B(1, 5, -4), and C(-2, 4, -6).

The midpoints of the sides can be found as follows:

- Midpoint of AB: MAB = ((2 + 1)/2, (1 + 5)/2, (-1 - 4)/2) = (1.5, 3, -2.5)
- Midpoint of BC: MBC = ((1 - 2)/2, (5 + 4)/2, (-4 - 6)/2) = (-0.5, 4.5, -5)
- Midpoint of AC: MAC = ((2 - 2)/2, (1 + 4)/2, (-1 - 6)/2) = (0, 2.5, -3.5)

Step 2: Finding the Perpendicular Bisectors

Next, we need to find the equations of the perpendicular bisectors of the sides. The perpendicular bisector of a line segment is a line that is perpendicular to the segment and passes through its midpoint.

To find the equation of the perpendicular bisector of AB, we need to find its slope and then determine the perpendicular slope. The slope of AB can be found using the formula:

mAB = (y2 - y1)/(x2 - x1) = (5 - 1)/(1 - 2) = 4/(-1) = -4

The slope of the perpendicular bisector of AB is the negative reciprocal of the slope of AB. So, the slope of the perpendicular bisector is mperpAB = 1/4.

Using the slope-intercept form of a line (y = mx + b), we can substitute the midpoint MAB and the slope mperpAB to find the equation of the perpendicular bisector of AB:

3 = (1/4)(1.5) + b
3 = 3/8 + b
b = 3 - 3/8
b = 24/8 - 3/8
b = 21/8

Therefore, the equation of the perpendicular bisector of AB is y = (1/4)x + 21/8.

Similarly, we can find the equations of the perpendicular bisectors of BC and AC.

The equation of the perpendicular bisector of BC: y = (-2/9)x + 41/9
The equation of the perpendicular bisector of AC: y = (7/2)x/3 + 5/2

Step 3: Finding the Intersection