00:01
We are going to find the equation of the sphere which passes through the points 1, negative 3, 4, 1, negative 5, 2, and 1, negative 3, 0, and whose center is on the plane x plus y plus z equals 0.
00:19
So let's give names to the points through which passes the sphere.
00:25
This first point, let me write it down here better.
00:30
So the first point one at the three four let's call it p 1 p 2 will be this next point here 1 negative 5 2 and p 3 is point 1 negative 3 0 also we're gonna to say that the point in the space x0, y0, z0 is the center of the sphere, center of the sphere, and so the information that the center is on the plane x plus y plus z equals zero means that the sum of the coordinates of the center is equal to zero.
01:36
So that's a condition we know x0 plus y0 plus z0 gotta be equal to 0 let's say this condition here we're going to call equation 1 good so now we're going to use the properties of the sphere the first one is that the distance from the center x0 y0 z0 to any point on the sphere gotta be equal to the radius it's it's a constant distance.
02:11
So distance from the center to, indeed, let me give it a name because i will need it to simplify notation.
02:23
So let, i'm gonna say it here, let c equal x zero y zero, c zero be the center of the sphere.
02:35
Okay, so we know the distance from the center of the sphere to any of these three points, to be 1, p2, p3 get to be equal to the radius that is the three distances are equal.
02:47
So we can say that by definition of the sphere we have that the distance from the center to p1 get to be equal to the distance from the center to p2 and that equal also to distance from the center to p 3 and is equal to the radius of the sphere and we're going to call that radius r so the radius r of the sphere okay so we're going to use this equality and this equality to get some information about the center of the sphere if the distance from the center of the sphere to p1 get to be equal to the distance from the center to p2 because we know the distance uses the square root we can say that this equality implies that the distance from the center to p1 square get to be equal to the square of the distance from c to p2 and so that means that the sum of the squares of the difference of the corresponding coordinates get to be be equal for points p1 and p2 with respect to the center.
04:39
So this distance here, this square of distance from c to p1 would be first coordinate of c is x0 minus first coordinate of p1 is 1 right here.
04:53
So x0 minus 1 squared plus second coordinate of the center is y0 minus the second coordinate of p1 is negative 3, so that becomes plus 3, and that square plus third coordinate of the centers is z0 minus third coordinate of p1 is 4, z0 minus 4 squared.
05:25
So this sum here is just the square of the distance from the center of the sphere to p1, and that gets to be equal to this square of the distance from c to p2 that is x zero minus first component of p2 let's see is one square plus second component of the center y zero minus second components of p2 is negative five so it has become plus square plus zero minus okay let me arrange this z zero minus and the third component of p2 is 2 in that difference squared.
06:13
So we can cancel out this x0 minus 1 squared, which is both sides of the equation, and we end up with y0 plus 3 squared plus c0 minus 4 squared equal y0 plus 5 squared plus z0 minus 2 squared.
06:43
And now we develop the squares here so we get y zero square plus six y zero plus nine and then plus z zero minus eight z zero plus sixteen that's equal to y zero square plus 10 y zero plus the square of five is 25 plus zero square minus 4 z 0 plus 4 good so now terms that cancel out at y 0 square and z 0 square so on the left we have 6 y 0 minus 8 z 0 and then this 10 y 0 on the right put it to the left you get negative 10 y 0 and this negative 4 z 0 we put it to the left and we get plus 4 is 0 and that is equal to on the right we have the constant 25 plus 4 and the constant on the left put it to the right is negative 9 and this one will be negative 16 so 6 y 0 minus 10 y 0 is negative 4 y 0 and negative 8 is 0 plus 4 is 0 is negative 4 z 0 equal to 2, negative 9 minus 60 is negative 25, that cancels out with 25, so we get 4.
08:27
And if we divide by negative 4 both sides of the equation, we get y0 plus z0 equal negative 1, and from here we can say that z0, for example, is negative 1 minus y0.
08:47
And so we get equation number 2 here.
08:51
Good.
08:52
Good.
08:53
So now we're going to do the same we did here, but now using these two other distances here.
09:03
The distance from the center of the sphere to p2 is the same distance as the center of the sphere to p3.
09:12
So let's see that.
09:16
So let's call it number two and that is distance from center to p 2 equal the distance from the center to p 3 implies that the squares are equal that is the distance from the center to p 2 square that would be equal to this distance from the center to p 3 square and now we write down these squares here so we get get x0 minus, let me see, p2 now here, so p2 is 1 square plus y0 plus 5 square, because second component of p2 is negative 5, and then 2, that is plus z0 minus 2 square, that is equal to x0 minus the first component of p3 is 1, so we get minus 1, squared plus y0 minus second component of p3, so you add to 3, so this becomes plus 3 squared, plus z0 minus 0, is the third component of p3, you see here, and that difference squared, so we have this equality here...