Although the path x : [0, 2] R2, x(t) = (cost,sin t) may be the most familiar way to

Chapter 3, Problem 34

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Although the path x : [0, 2] R2, x(t) = (cost,sin t) may be the most familiar way to give a parametric description of a unit circle, in this problem you will develop a different set of parametric equations that gives the x- and y-coordinates of a point on the circle in terms of rational functions of the parameter. (This particular parametrization turns out to be useful in the branch of mathematics known as number theory.) To set things up, begin with the unit circle x 2 + y2 = 1 and consider all lines through the point (1, 0). (See Figure 3.15.) Note that every line other than the vertical line x = 1 intersects the circle at a point (x, y) other than (1, 0). Let the parameter t be the slope of the line joining (1, 0) and a point (x, y) on the circle. (1, 0 ) (x, y) y x Slope t Figure 3.15 Figure for Exercise 34. (a) Give an equation for the line of slope t joining (1, 0) and (x, y). (Your answer should involve x, y, and t.) (b) Use your answer in part (a) to write y in terms of x and t. Then substitute this expression for y into the equation for the unit circle. Solve the resulting equations for x in terms of t. Your answer(s) for x will give the points of intersection of the line and the circle. (c) Use your result in part (b) to give a set of parametric equations for points (x, y) on the unit circle. (d) Does your parametrization in part (c) cover the entire circle? Which, if any, points are missed?