Let x: (0,π) → R2 be the path given by where t is the angle that the...

Let x: (0,π) → R² be the path given by

where t is the angle that the y-axis makes with the vector x(t). The image of x is called the tractrix. (See Figure 3.49.)

(a) Show that x has nonzero speed except when t = π/2.

(b) Show that the length of the segment of the tangent to the tractrix between the point of tangency and the y-axis is always equal to 1. This means that the image curve has the following description: Let a horse pull a heavy load by a rope of length 1.

Suppose that the horse initially is at (0, 0), the load at (1, 0), and let the horse walk along the y-axis. The load follows the image of the tractrix.