00:01
In this problem, we want to find an equation for the surface consisting of all points that are equidistant from the point minus 3 -0 -0 and the plane x equal to 3.
00:15
So here i've drawn a 3d cartesian grid, and i've labeled our point in red minus 3 -0, and in blue we have our plane located at x is equal to 3.
00:30
Next, let's consider an arbitrary point with coordinates x, y, z.
00:39
This coordinate is going to be on our surface if the distance between this coordinate and our point minus 3 .00 is equal to the distance to our arbitrary point x, y, z to our plane.
00:58
I'm going to call this first distance, the distance between our arbitrary point x, y, z, to our fixed point minus 300 d1 i'm going to call the second distance d2 the distance between our arbitrary point x y z to our plane x equal to 3 what we're going to do is we're going to characterize d1 in d2 and equate them to find our surface so let's start by characterizing the points or rather sorry the distance between our arbitrary point x, y, z, and our fixed point minus 3 0.
01:50
So this distance i will label it d1 and d1 squared is going to be equal to x minus 3 squared plus y minus 0 squared plus z minus 0 squared.
02:08
Which simplifies to x plus 3 square plus which can correct this plus y square plus z square this is our first distance d1 that we've just found an expression for now let's find an expression for our distance between our arbitrary point x y z and our plane a point on our plane will have coordinate 3 y z let's find d2 squared d2 squared is going to be equal to x minus 3 squared plus y minus y squared plus z minus z squared which is equal to x minus 3 square great now we have expression for d1 and d2 so now let's equate d1 and d2.
03:31
And since we have everything squared, let's simply equate d1 squared and d2 squared.
03:36
Why do you want to equate these two? because the points, x, y, z that satisfy that equation, will form our surface.
03:46
So d1 squared is equal to x plus 3 squared plus y square plus z square plus z square.
03:57
Is going to be equal to x minus 3 squared.
04:03
Let's develop x plus 3 squared on our left hand side and x minus 3 squared on our right hand side, we're going to find that x squared plus 9 plus 6x plus y square plus z square is equal to x square plus 9 minus 6x.
04:28
Now we see that on our left hand side and right hand side, we will have x square plus 9 that will cancel out.
04:38
So our equation will simplify to y square plus z square is equal to minus 12x.
04:51
We're going to rewrite this equation as x is equal to minus y square over 12 plus z square over 12.
05:08
Great, so now we've just found a relation that characterizes our surface consisting of points that are equidistant to our point minus 3 00 and to our plane x equal to 3.
05:25
So what i want to do is determine what is this the surface...