The paraboloid z=x^2+y^2 and the plane z=2 x+4 y+4 intersect...

The paraboloid $z=x^{2}+y^{2}$ and the plane $z=2 x+4 y+4$ intersect in a curve in 3 -space. (a) Show that the shadow of the intersection in the $x y-$ plane is a circle and find its center and radius. (b) Parameterize the circle in the $x y$ -plane. (c) Parameterize the intersection of the paraboloid and the plane in 3 -space.

The paraboloid $z=x^{2}+y^{2}$ and the plane $z=2 x+4 y+4$ intersect in a curve in 3 -space.
(a) Show that the shadow of the intersection in the $x y-$ plane is a circle and find its center and radius.
(b) Parameterize the circle in the $x y$ -plane.
(c) Parameterize the intersection of the paraboloid and the plane in 3 -space.

Key Concepts

Projection onto the Coordinate Plane

Projection refers to the mapping of a higher-dimensional curve or set onto a lower-dimensional plane by ignoring one or more coordinates. Here, the 'shadow' of the intersection on the xy-plane is obtained by projecting the three-dimensional curve onto two dimensions. Recognizing the geometric shape of this projection, particularly its circle shape, is an important aspect of the problem.

Parameterization of Intersection Curves in 3D

After parameterizing the two-dimensional curve, the next step is to lift this parameterization into three dimensions to obtain the curve of intersection between the two surfaces. This requires substituting the parameterized x and y coordinates into one of the original surface equations to obtain the corresponding z coordinate, thus giving a complete three-dimensional parameterization of the intersection curve.

Parameterization of a Circle

Parameterization is the process of expressing the points of a curve using a single parameter, typically using trigonometric functions for circles. This involves writing the x and y coordinates in terms of sine and cosine functions, thereby capturing the circular nature of the projection in a concise form. This representation is very useful for further analysis or integration in problems involving circles.

Intersection of Surfaces

This concept involves finding the points that simultaneously satisfy the equations defining two surfaces in three-dimensional space. The result of this intersection is typically a curve or set of points that lies on both surfaces. In the problem, the intersection of the paraboloid and the plane results in a space curve, and analyzing it requires understanding how to set the two equations equal and manipulate them accordingly.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic equations into a form that explicitly reveals the geometric characteristics of conic sections, such as circles. By rearranging and grouping terms, one can identify the center and radius of a circle directly. This method is key in transforming the equation derived from the intersection into the standard form of a circle.

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