Undergraduate Research Project Find the polar equation of the envelope of circles in the e...

Undergraduate Research Project Find the polar equation of the envelope of circles in the experiment of Problem 6 using polar coordinates and the following note.

NOTE: The subject of envelopes generated by a one-parameter system of curves F(x,y,t) = 0 is sometimes found in the older books on differential equations. The main result is the following (under certain hypotheses): The envelope of f(x. y, t) = 0 is given by the solution of the following system of equations in x and y. where t is eliminated algebraically:

You should reference this result as part of the project. These equations should be converted to polar coordinates in order to work Problem 6, as mentioned. (Use the conversion equations x = rcosθ, y = rsinθ; the partial derivative is not affected since the differentiation is with respect to t.)

Example: Let F(x, y,t) = x/t + ry - 1 (family of lines with x- and y- intercepts t and 1/t respectively, which will generate a hyperbola). The system

{∂F/∂t = x{-t-2) + y = 0, F = x/t + ty -1=0} reduces to 4xy = 1.