(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find an equation of the form Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0 Satisfied by the set of all points in space, (x,y,z), whose distance to the origin is equal to their distance to the plane x+y+z=3. Based on what you know about parabolas, what does this collection of points look like?

2. Relevant equations

x+y+z=3

Equation of a paraboloid: z/c=x^{2}/a^{2}+y^{2}/b^{2}

a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0 The coefficients (a,b,c) is the normal vector to the plane.

3. The attempt at a solution

I started by finding a point that lies on the plane. The point (1,1,1) satisfies the given equation: x+y+z=3.

Given that point, I can work back to the normal vector:

(x-1)+(y-1)+(z-1)=0

The normal vector is the coefficients of this eqn, so the normal vector is <1,1,1>.

The focus is given as (0,0,0), so the vertex of the paraboloid should be [tex]\frac{1}{2},\frac{1}{2},\frac{1}{2}[/tex]

The distance from the origin (and thus the plane) to the vertex of the paraboloid is 1/2[tex]\left\|N\right\|[/tex]=[tex]\frac{\sqrt{3}}{2}[/tex]

That's as far as I've gotten... I really have no idea how to go from here. I think I've got all the info to put it together, I just don't know how.

BTW, sorry about any formatting snafus, Google Chrome doesn't play well with latex at all.

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# Homework Help: Finding the equation of a paraboloid

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