Difficult multivariable problem to find equation for 3d surface

[Jimmy5050]

Homework Statement

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.

Homework 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.

The Attempt at a Solution

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

 

[vela]

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)?

 

[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!

 

[vela]

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!

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?

 

[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"?

 

[vela]

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:

D1= abs(ax+by+cz+d)/sqrt(a^2+b^2+c^2)

In your next step, it was gone, so I assumed you set it to 0.

 

[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!

 

[vela]

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

 

[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?

 

[vela]

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|.