14. A circle of radius a is placed inside a larger circle of radius b. Initially,...

 

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14. A circle of radius a is placed inside a larger circle of radius b. Initially, the small circle is at rest at the bottom of the large circle. Let's set the origin of a Cartesian coordinate system to be the point where these two circles touch, with the centers of both circles on the y-axis. Now, we roll the little circle counter-clockwise inside the big circle, and trace the path of the point (x, y) on the little circle that was initially touching the big circle. a. If (in radians) is the angle between the center of the small circle and it's initial location, find a parametric equation for x and y in terms of 0. b. What relationship must be satisfied between a and b for the point to end up where it started after exactly one roll around the big circle? c. Show that if b/a 2, then the curve traced by any point on the inner circle is a straight line. 14. A circle of radius a is placed inside a larger circle of radius b. Initially, the small circle is at rest at the bottom of the large circle. Let's set the origin of a Cartesian coordinate system to be the point where these two circles touch, with the centers of both circles on the y-axis. Now, we roll the little circle counter-clockwise inside the big circle, and trace the path of the point (x, y) on the little circle that was initially touching the big circle. a. If (in radians) is the angle between the center of the small circle and it's initial location, find a parametric equation for x and y in terms of 0. b. What relationship must be satisfied between a and b for the point to end up where it started after exactly one roll around the big circle? c. Show that if b/a 2, then the curve traced by any point on the inner circle is a straight line.