- #1
Quisquis
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Homework Statement
Find an equation of the form Ax2+By2+Cz2+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?
Homework Equations
x+y+z=3
Equation of a paraboloid: z/c=x2/a2+y2/b2
a(x-x0)+b(y-y0)+c(z-z0)=0 The coefficients (a,b,c) is the normal vector to the plane.
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.