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# Find an equation for the plane consisting of all points that are equidistant from the points $(2, 5, 5)$ and $(-6, 3, 1)$.

## $4 x+y+2 z=2$

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

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.

DM
Oklahoma State University

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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