Parametrization of Intersection using Polar Coordinates

Parametrize the intersection of the cylinder $X^2 + Y^2 = 4$ and the plane $2X + 3Y + 5Z = 10$ using polar coordinates for X and Y, and solving for Z. Use $R(T) = (2\cos(T), 2\sin(T), 2 - \frac{4}{5}\cos(T) - \frac{6}{5}\sin(T))$ for T in $[0, 2\pi]$.

In tackling the problem of finding the parametrization for the intersection of a cylinder and a plane, we delve into the concepts of polar coordinates and parametric equations. The given cylinder is defined by the equation $X^2 + Y^2 = 4$. In cylindrical coordinates, each point on a circle can be represented using radius and angle, simplifying the process of parametrization in the XY-plane. By leveraging this method, we can express both X and Y as functions of the angle T, which ranges from 0 to 2π, capturing the complete circular path.

Furthermore, this problem requires us to find the Z value for each point on the path of intersection. The plane is described by a linear equation in terms of X, Y, and Z. By substituting the expressions for X and Y from the polar coordinates into this plane equation, we can solve for Z as a function of T. This completes the parametrization, giving us a full description of the curve of intersection in three dimensions using the parameter T.

This task not only exercises your understanding of polar coordinates as a tool for parametrizing curves in a plane but also extends these concepts into three-dimensional space. It highlights the importance of converting between different methods of coordinate representation to simplify complex geometric intersections, a common theme in multivariable calculus and analytic geometry.

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