Multivariable Calculus~Equation of a Sphere

[madisonfly]

Homework Statement

Find the Equation of the sphere with points P such that the distance from P to A is twice the distance from P to B.

A(-2, 4, 2), B(4, 3, -1)

Homework Equations

The equation of a sphere would probably be the most relevant equation.

That is (x-h)^2 + (y-k)^2 +(z-l)^2 = r^2

The Attempt at a Solution

So the way I look at it, I figure that I have to set up an equality. Therefore d(PA) = 2d(PB). I'm assuming that since point p isn't given, it is P(x, y, z)? I don't know though. If that's the case, my equation should look something like

(x-2)^2 +(y-4)^2 +(z-2)^2 = 2((x-4)^2 +(y-3)^2 +(z+1)^2). But I'm not sure if I'm approaching the problem the right way. And I don't where to look for the radius. Any input would be appreciated! :)

 

[rock.freak667]

The Attempt at a Solution

So the way I look at it, I figure that I have to set up an equality. Therefore d(PA) = 2d(PB). I'm assuming that since point p isn't given, it is P(x, y, z)? I don't know though. If that's the case, my equation should look something like

(x-2)^2 +(y-4)^2 +(z-2)^2 = 2((x-4)^2 +(y-3)^2 +(z+1)^2). But I'm not sure if I'm approaching the problem the right way. And I don't where to look for the radius. Any input would be appreciated! :)

I believe you are approaching it correctly, but that '2' would be converted to a '4' when you square both sides.

Distance =√[(x-x₁)²+(y-y₁)²+(z-z₁)²]


So just expand out your equation, collect the like terms and then complete the square for each variable again.

 

[madisonfly]

Ohhhh. Nice thanks a bunch! I hate stupid numerical mistakes like that.