3. The goal of this problem is to find a parametrization for the intersection of the paraboloid z= x^2 + y^2 and the plane 4x − 2y − z = −25. (a) Draw a picture of a paraboloid intersecting a plane. What possibilities are there for the shape of the intersection? (b) Solve the second equation for z and substitute this into the first equation. In the plane, the resulting equation describes a circle. Find the center and radius of this circle. (c) Find a parametrization of the circle from part (b). (d) Use your answer to (c) to write z in terms of t.
3. The goal of this problem is to find a parametrization for the intersection of the paraboloid z= x^2 + y^2 and the plane 4x − 2y − z = −25.
(a) Draw a picture of a paraboloid intersecting a plane. What possibilities are there for the shape of the intersection?
(b) Solve the second equation for z and substitute this into the first equation. In the plane, the resulting equation describes a circle. Find the center and radius of this circle.
(c) Find a parametrization of the circle from part (b).
(d) Use your answer to (c) to write z in terms of t.
(e) Using your answers to parts (c) and (d), write a vector parametrization of the intersection between the paraboloid and the plane.
(f) Consider again your answer to part (a). Is your answer reasonable?
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