Consider the points $ P $ such that the distance from $ P $ to $ A (-1, 5, 3) $ is twice the distance from $ P $ to $ B (6, 2, -2) $. Show that the set of all such points is a sphere, and find its center and radius.
all right, So this problem is asking us to consider the point p such that the business from P. T. A right here is twice a distance from P to be. Now it says to show that the set of all of these points is a sphere and to fight at center and radius. Okay, so we've learned that if we want to define some vector p a, the length of that victor is going to be equal to essentially, you'll take the square root. Then you'll take the X will call the coordinates of P just X, y and Z. So we'll say escorted a p minus. The escort of Ace doesn't minus one settled into being a plus one. It's weird. Waas Why than minus five? The white corner of a plus Z minus three, which is the Wycoff Zeke wanted of a parties. This is, like pretty familiar. So what? Like the Pythagorean theorem, that very scene ever going to find the vector between any point p and B the length of that of those measures. Same shorts up to get X minus six squared. Plus why minus two squared plus Z minus negative, too. So we plus two squared. I'm sorry about the horrible handwriting. It's not ideal. Eso This problem is saying that all these points p the length between the distance between them and they will be twice isn't between them and be What's that saying? Oh, this length p a is twice the length P b. Right. And remember, we're interested in showing that all these points p form some sort of sphere. Right? So we have these two equations, um, and we're going to plug them in to this formula here, right of a plug him in like this will have the square root signs s. So it seems like the first thing that we should try to do I will be the square, both sides of this equation. So we're gonna do it. A second is we're going to take this side. Ooh, and square it. It's not what we do right there. I'll take this side and square it. I don't notice that when you square that, uh, this side of the equation, we'll actually get a four instead of a two there. Okay, so I'm gonna refresh his page. Um, just keep buying that That's what we're doing. We're gonna lose all their stuff, though. Okay, so we're going to get X Plus one squared. Plus why minus five squared plus Z minus three. That's minus. It's weird. Equals. Now we have a four, Remember? Because we square both sides. Ah, we're gonna put Brackett. You might have to go down the line for this. Might bring a room X minus six, I swear. Plus, why minus two squared plus Z plus two squared. All right, so now the next thing we have to do, um, try to figure out what we can make with this, we're goingto have to actually square all these terms. Right? Um, so this is going to be a very long drawn equation, so I might have to refresh the page again. We might have to lose this, but So, for example, this thing is going to turn into X squared. Plus two x, uh, plus one. This thing will be, you know what was right it out. It's what we're doing right now is actually getting rid of all those squares, right? Two x plus one for that 1st 1 Plus that we're under a second term of the p A. But equation my scream eyes 10 wide Plus who is a Why? Sorry about that is an ugly why plus 25 plus z squared minus 60 plus nine. So this is all just the p A squared length of p A squared? Um Then we actually did the work on the binomial sze That's going to be able to four times when do the same thing for the peopie equation. So X weird mice 12 x plus 36 plus why squared mice for why? Plus four plus z squared plus for Z who is liable for plus for right. So we worked out of the binomial. Lt's now we're going to distribute this four into all of these. Um so what is try Do this in the next line. So once we distribute for I should give us four x squared minus 48 x plus Ah, 144 plus for yeah, Why squared by a 16? Why? Plus 16? That's a really bad plus. I'm sorry all around A tough day Plus for Z squared plus 16 z plus 16. Okay, so now the next you're going to do we're still remember Our endgame is still to try to get this into something that looks like a satirical equation. Right? So what you have and one of those equations is you have, like, an X minus a squared. Plus, why Might is B squared plus C minus C squared equal. Some are squared. Right? Um, that's the former look going for. So what we're gonna do right now is we're going to get all of our constance on one side. Um, and then we're going to get over variables on the other side, okay? And then we'll try to complete the square on and see where that leads us. So I wrote down the math on this. Um, according to my numbers, if we put all the constants are one side of these two equations, we're going to get negative 141. Then remember, we had a bunch of we had a bunch of ex squares y squares and Z squared with fours in front of them on the right side of the equation, and one's on the other side. So if we subtract, all the wines were going to get a bunch of threes for those. So we're going to get three x squared and we subtracted. We ended up with minus 50 X plus three. Why squared minus six. Why? Plus three z squared. Plus 22. See? All right, so we're starting to make progress here. Right. Um So what we need to do to get us to our spirit will form is complete these squares, right. So he at a constant here, that'll make this a square binomial. Right. Um, so if you remember how by how the squares work. Well, first of all, we're going to buy this all by three because we don't want coefficients in front of our actual variables in the end. Um, so when we divide by three here, we'll write that out. Going to get negative. 47 here equals X squared mice. 53rd x plus. Why squared minus two. Why? Plus z squared. Plus 22 over three. See, now, remember the nature of Ah, excuse me of a binomial. What we're gonna have is the constant that we need to add teach of these things. So to complete the square for this X one, for instance, we're going to need to add whatever is half of this coefficient here. I'm going to need to add my apologies. My bed. Okay, so the final term we're going to get here is going to be ex minus half the confident We're going to get exercise. 50 over six squared as a binomial here, Um, which means that we need to add 50 over six squared right to complete the square. We're going to keep doing this. So eventually going to get why minus one cure. Which means we're gonna have to add a one. We're going to get Z plus 22 over six here, which means we need to add 22 over six squared to this equation. So what we're gonna do is we're going to add Ah, 50 over six squared, one squared than 22 over six squared. They're going to Adam to both sides. So plus, Oh, my gosh. Plus, it's not wanting to write. Plus all these numbers, you know, you get where I'm going with this squared, we're running out of space, but we're going to work. It's one squared. Plus, uh, well, we've definitely run the space, but you get the image right. We're going to add these Constance to both sides of the equation. and those are some wacky numbers. Luckily for you, I already figured out what they worked out, too. Um, so but we end up with right. Okay. Is this equation where we have Come on? If we simple fire fractions, we get X minus 25 over three. That's west early pregnancy swearing. Plus, why minus one squared plus Z plus 11 3rd squared, and then we're going to get this crazy number on the other side. Um, it's going to be equal to negative 47 plus all this conscience that we had to add to completely squares, right, So that's gonna be negative. 47. Plus, all those things is going to get us to 332 over nine. Okay, So that means that our center here because it asks. This is a sphere, right? Just crazy. Got all the way down to this spherical equation through all that. Um, So our center here is going to be, you know, the form is going to be 25 over three one, the negative 11 over three as her center. Then our radius is going to be the square root of this square. Root 332 over nine. Make that bigger. There you have it. That's how you saw that equation.