Find an equation for the plane consisting of all points that are equidistant from the points $ (2, 5, 5) $ and $ (-6, 3, 1) $.
$4 x+y+2 z=2$
So in this problem we are asked to find an equation for the plane consisting of all the points that are equal distance From these two points 255 and -631. So if we let this be point A and this be point B. Then the equation for these points in this plane, we'll call those points P x Y z right X y z coordinates to him. And we know that the distance uh from P to a physical, the distance from P to be. So then we also know that P minus a squared is equal to p minus B squared as well. So what we can do is we can take in the X direction have x minus two squared Plus in the Y Direction Y -5 squared. And in the z direction z minus five squared is going to be equal to x minus of minus six plus six plus six squared for us, Y -3 Squared plus Z -1 Squared. So multiply all this out. Okay. And what do we get? Well we have x square gonna show up here and over here. So those are going to cancel out. So we're gonna be left with a minus four X plus four over here And a six x plus 36 over here. Okay, I'm sorry, none of six x oops. One second. Right. That would be a 12 X plus 36. And then why squared? And y squared? There's cancel out? So we're gonna have minus 10 Y plus 25 left over here. We're going to have minus six Y plus nine left over there. And Z squared and Z squared over here. Going to cancel out. So we're going to be left with minus 10 Z plus 25 And -2 Z plus one left over there. Move everything to one side. And what happens? Well Yeah we subtract 12 x there. All right. And we'll have -16 x. For those terms add six Y to both sides. That that's minus for why resulting? And then add to Z. That That's -8 Z. And then what do I have? I have 25 and 25 is 50 And four. That's 54. And I must subtract 36. So that leaves 18 straight nine more. That leaves me nine and one more. So many lives he left with plus eight equals 0 Notice there's a common factor of -4 here. So this is for ex plus why? Plus two Z minus two equals zero. And there is the equation of the plane of all the points equal distance From those two points A&B.