Difficult multivariable problem to find equation for 3d surface

1. Sep 18, 2010

Jimmy5050

1. The problem statement, all variables and given/known data

A given surface contains all points G such that the distance from G to the plane z=4 is double the distance from point G to the pt. (2, -3, 1). Find eqn for the surface.

2. Relevant equations

I thought the distance formula for a point to a plane would help, but I can tell that the equation is going to have to be that of a parabola.

3. The attempt at a solution

I've made at least 5 unsuccessful attempts, and have NO idea where to go from here.

2. Sep 19, 2010

vela

Staff Emeritus
Say the point G has coordinate (x, y, z). Can you get expressions for the distance between this point and the plane z=4 and for the distance between this point and (2, -3, 1)?

3. Sep 20, 2010

Jimmy5050

For the distance formula between arbitrary point (x,y,z) on the surface to the plane z=4 would be;

D1= abs(ax+by+cz+d)/sqrt(a^2+b^2+c^2) where n=<a,b,c> is the normal vector to the plane z=4, which we can say is <0,0,1>.

So D1=abs(0x+0x+1z)/sqrt(0^2+0^2+1^2) = abs(z)

And the distance from the point G (x,y,z) and the point (2,-3,1) is

D2=sqrt( (x-2)^2 + (y+3)^2 + (z-1)^2 )

So 2D1=D2

So 2z=sqrt( (x-2)^2 + (y+3)^2 + (z-1)^2 )

I feel like something is wrong however, THANKS for the help!

4. Sep 20, 2010

vela

Staff Emeritus
This is almost right, but you got the distance between the point and the plane z=4 wrong because you set d=0. The origin, for example, is a distance 4 away from the plane, but |z|=0. Similarly, the point (0,0,4) is on the plane, so there's 0 distance between it and the plane, not a distance of |z|=4. Can you see how to fix your answer?

5. Sep 20, 2010

Jimmy5050

I'm still not understanding what I have to correct.

I guess I don't really understand what should be plugged in for d. From my understanding, d is going to have to be some type of function because it is changing as the surface changes to different points "G"?

6. Sep 20, 2010

vela

Staff Emeritus
Your formula for D2 is correct. Your formula for D1, however, isn't. I gave you two examples where it clearly gives the wrong answer.

When I said d=0, I was referring to the d which appears here:
In your next step, it was gone, so I assumed you set it to 0.

7. Sep 20, 2010

Jimmy5050

Ok... Since the distance from the point to the origin is d, then we can say that the distance from the point to the plane would be z-4

So in the equation, d = z-4

Thanks for all the help!

8. Sep 20, 2010

vela

Staff Emeritus
Is that d or what you called D1 earlier? Remember d is a constant in that formula; z shouldn't appear in it.

9. Sep 20, 2010

Jimmy5050

I meant it as d, not D1.

I now realize its a constant, so solving through with that same logic should just give d= -4?

10. Sep 20, 2010

vela

Staff Emeritus
Yes. If you rewrite the equation of the plane as z-4 = 0, so that it's in the form of ax+by+cz+d=0, the -4 is d, so D1=|z-4|.