Problem 45 Easy Difficulty

Find an equation of the set of all points equidistant from the points $ A (-1, 5, 3) $ and $ B (6, 2, -2) $. Describe the set.

Answer

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Video Transcript

Let's find the equation that describes all points that are equidistant between points, a negative 153 and B six too negative, too. What's that gonna look like? Well, if we're in two dimensions and I have two points, we'll just call this one A and B the points that are the same distance between A and B fall along this line, which is the perpendicular by sector of the line segment from A to B and all of the points on this line where the same distance from A as they are to be and three dimensions we can imagine viewing this line from many perspectives. Imagine holding down points A and B and spinning it around, coming in and out of the writing surface along that access. That's the line segment between them that's going to give you a whole plane of lines that are equal distant between A and B. So in three dimensions we're looking at a plane. All of these lines fallen. So said any point it's call it P on that plane. You mean the same distance away from A as it is from B A pre equals BP, and that's gonna be the basis for coming up with our equation. We're going to assign P coordinates X, Y and Z because it could be any point on this plane. So whatever point we're referring to is gonna have coordinates X, Y and Z. We're gonna find the distance from HP using the distance formula, find the distance from B to pee and set them equal to each other. And that's going to give us the equation that describes all of these points that are the same distance from A as they are to be. So supply the distance formula using X, Y and Z four point p and the coordinates negative 153 for point A and six to negative two for point B, so the distance from okay to pee is the square root of subtracting. X coordinates X minus negative one square plus Why minus five square plus Z minus three. Square the distance from B to pee. It's attracting ex cornets X minus six square plus y minus two square plus Z minus negative too square. I'm going to observe that if a P is equal to be pain, then there squares must be equal to each other. and I want to do that to kind of simplify the outs with them. About to do that will get rid of my square roots, like look at the squares of these distances and set them equal to each other. We're not gonna have to worry about any extraneous roots being introduced because the radicals inside here can never be negative. So we're not gonna have the possibility that one was negative and one was positive at a new solutions. So let's square each of these, which basically removes a radical, set the medical to each other and then do some algebra to simplify everything. So I'm gonna have X plus one swear plus why minus five square plus Z minus three square and the distance from B to P square on the right hand side, X minus six square plus why minus two squared plus Z plus two square and I will multiply these out and you can do this either using foil or I'm going to use wearing and by no meal. A plus fee square is equal to any square, plus to a B. Let's be squared. Use this formula, and when I do this in this example, I'm gonna do this middle multiplication fairly quickly. So if I go too fast for envies, you can stop the video real quick. Do the I was a little bit more slowly either perfect square polynomial or just foiling it out. And then once you're convinced you got the same, try no meal for emergencies. You can go ahead and continue with the video. I'm going to have X square These X squared plus two x plus one plus y squared minus 10. Why I was 25 plus Z square minus six z plus nine equals X squared minus 12 acts was 36 plus y squared minus four. Why plus four plus the square plus four z plus four. There were a lot of terms there, but many of them are about to cancel out cause I have X squared on both sides of the equation. Why? I swear on both sides of this equation and Z square on both sides of this equation, the only like terms to collect are the constants. And so, if I take X squared plus y squared plus B squared, it's attract that on both sides. The X squares we're gonna cancel. But why? Squares are gonna cancel and the C Square's we're gonna cancel. And that's just going to give me a linear equation, linear terms on each side. So as I write this, I will write the variable terms and then I'll combine the constants on each side, and then we'll switch sides but the variables on one side and the constants on the other. But let's not do that stuff yet. So on the left hand side we're left with two acts minus 10. Why minus 60 plus when an one plus 25 is 26 Let's nine is 35. So that's how the left hand side simplifies. On the right hand side, I have negative 12 acts minus four. Why plus four z plus and I'm going to add the 36 plus four this 40 plus four. This 44. Now we're all of the variables to the left hand side, with the constant to the right hand side, and that will give us the form of the equation up. Best gives us information about what the object looks like. Someone has add 12 x to both sides. 12 X plus two X is 14 axe I'm going to add. For why? To both sides. Negative 10. Why? Plus for why is gonna give me negative six. Why? I'm going to subtract for Z from both sides. So when have negative six Z minus four Z It's gonna give me negative 10 z equals it was Subtract 35 from both sides to get the constant term on the opposite side and 44 minus 35 is nine. And so my equation for the plane of all points that are the same distance from A as they are to be is 14 x minus six. Why minus 10 z equals nine.