00:05
Okay, so we want to solve this problem here.
00:09
So we want to consider the points p such that the distance from p to a is twice the distance from p to b.
00:17
And we want to show that the set of points is a sphere and find the center and radius.
00:24
So the first thing is let's write out the data we have.
00:27
So i'm going to call a set s.
00:30
This is going to be the set of all the points p.
00:34
Which we can, you know, we're in three -dimensional space, so we can like this as x, y, z, and r3, such that on the distance from p to a to be two times the distance from p to b.
01:02
So this is the set we're dealing with, and we can show that this is a sphere.
01:07
So let's write our goal to show s is a sphere.
01:21
So first things we want to do is, well, what does it mean to be a sphere? so let's just write that out just so we know what we're trying to get.
01:30
So a sphere is a set of points of the form.
01:37
So we want some set of points x, y, z, and r3, such that x minus a squared plus y minus b squared, add z minus c squared is equal to r squared for some a, b, c, and r, which are real numbers.
02:12
Okay, so if we look at the two things, the first, the bit before this line, they're the same.
02:19
So that's all good.
02:20
Now, what we have to do is make this information after the line, and we don't want to try and get it to be of this form down here.
02:29
So let's write out what we have.
02:32
So we know that a is the point minus one, five, three, third, b is the point six, two, two, minus two.
02:51
Okay.
02:52
So now what's the distance from p, which is this vector here? so it's, we, i mean, p's arbitrary, but we can write it in the form x, y, z.
03:01
So we want to know what is the distance from p to a.
03:07
So a distance from p to a.
03:09
What's this equal to? and we want to use the distance formula here.
03:13
So if remember the distance formula, the, so let's write it over here.
03:19
So if we've got two points p and q, then the distance from p to q is equal to square root of p1 minus q1 squared, add p2 minus q2 squared, add p3 minus q3 squared, okay, so we want to use this formula here to figure out an expression for the distance from a point p in the set to a.
03:51
So let's do this.
03:54
So this is going to be the square root of, so the first component of p, which is x, minus the first component of a.
04:01
So we have plus 1 squared.
04:04
And now for the second component, y minus 5 squared.
04:08
And the third component z minus 3 squared.
04:15
Okay, now we do the same thing for the point b.
04:22
So the distance from point p to b is going to be given by x minus 6 squared plus y minus 2 squared plus z plus 2 squared.
04:43
Okay, now we know from the question that the distance distance from p to a is two times the distance from p to b.
04:55
So this tells us we know this is from p to a two times it seems from p to b.
05:10
So using these formulas this tells us that the square root of, well, let's just save some time, let's copy this.
05:23
So we know that this is equal to two times two times.
05:31
Two times this.
05:42
Now, the good thing about distances is that they're all positive.
05:47
So because they're positive, this is the same thing, so this is true if and only if, and then we can just square both sides.
05:55
So this says that x plus 1 squared add y minus 5.
06:03
Make a bit clearer, y minus 5 squared, add z minus 3 squared is equal to 4 times x minus 6 squared plus 4 times y minus 2 squared plus 4 z plus 2 squared.
06:32
Okay, so i mean, so far all we've done is just pass this four in these two formulas and use our hypothesis that these have this equality here.
06:44
So what do we do now? well, our goal, if we remember, is we want to show the s's it's i want to enter this to get something.
06:58
So in particular, because we've got x's and y is appearing in multiple places, we want to group them together.
07:03
So the best way to do that is unfortunately we just have to expand everything.
07:08
So when we do this, we're going to get x squared plus two.
07:14
2x plus 1 add y squared minus 10y plus 25 plus z minus 6 z plus 9 and so that's the left -hand side this is going to be equal to now if my math is right 4x squared minus 48 x plus 144 we're going to know for the y term we're going to have 4 y squared minus 16 y plus 16 and 4 finally for the z 4 z squared add 16 z plus 16 okay so we've got this have long equation now what i want to do is so i'm going to move so let's just write out in red what i go what the next step is move variables.
08:38
So remove all the variables from here over to here.
08:42
And then i want to move constants.
08:51
And we're moving from this side over to this side.
08:54
Because you know, our goal is we want this, this, this expression here that's highlighted.
08:59
You know, all of the variables from one side and then we've got just the constant term on the right.
09:05
So that's kind of what we're trying to aim towards over here.
09:09
So moving stuff over here, i'm just going to flip the equal.
09:12
As well just to make things a bit neater.
09:17
So what do we end up with? so we've got we're going to move x squared over here so we're going to end up with 3x squared and let's just cancel them off once we've used them so we know what and what we have left.
09:37
3x squared and then we're going to have minus 2.
09:41
So we're going to have minus 50x...