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skyturnred
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Homework Statement
Find the points nearest and furthest from the origin on the intersection of a plane with a paraboloid.
Plane:
x+y+2z=30
Paraboloid:
z=x[itex]^{2}[/itex]+y[itex]^{2}[/itex]
Homework Equations
The Attempt at a Solution
Obviously the first step is to find the equation of the ellipse that is formed
I do that by plugging the equation of the paraboloid into the equation of the plane and get:
x+y+2x[itex]^{2}[/itex]+2y[itex]^{2}[/itex]=30
But my problem is that the equation above isn't really the equation I need. The equation above gives the shape of the ellipse *on the plane* but I need an equation for that shape on the original 3D Cartesian plane. How do I go about doing that?
Also, once I find said equation, would the following way to go about finding the max/min be correct?
Choose the equation of the ellipse as the constraint, choose f(x,y,z)=[itex]\sqrt{x^{2}+y^{2}+z^{2}}[/itex] and then just treat it like any straight forward max/min question?
Thanks