The class notes yesterday should have included some discussion of the “self-intersecting curve” problem. Here it is a day late. Suppose *x = x(t)* and *y = y(t)* are given; a *point of self-intersection* is said to occur when, for two *different* values of the parameter— and , say—one has the *same* point of ; in other words and .

In the example at hand, we have and . Assume that the parameter values and give a point of self-intersection for this curve. From the equation for *x* we see that then ; since we are assuming thqt , this implies that . Substituing in the equation for *y* gives

; factoring the LHS gives , the *opposite* of the RHS. Since a number equal to its own opposite is zero, we have RHS = 0; factoring gives ; since (because ), this implies that .

So our parametric curve intersects itself for the *t* values and . One easily obtains the desired values of to complete the exercise.

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