Name: Anonymous 2009-09-16 21:14
I've been staring at this problem for hours, but can't figure out how to begin. I understand the projection of the intersection on the xy-plane is a circle with center (-9/2,7/2), which means the equation of that intersection would start out with (x+9/2)^2+(y-7/2)^2=, but how do i find the radius of this circle? If I can just figure out how to find the radius and plug that into the circle's equation i'll be set.
Consider the paraboloid z = x^2 + y^2. The plane 9 x - 7 y + z - 10 = 0 cuts the paraboloid, its intersection being a curve.
Find "the natural" parametrization of this curve.
Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface.
Consider the paraboloid z = x^2 + y^2. The plane 9 x - 7 y + z - 10 = 0 cuts the paraboloid, its intersection being a curve.
Find "the natural" parametrization of this curve.
Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface.