Exercise 14.9. This exercise uses a linear transformation to relate the curve H+ defined by x2 - y? = +1 to the hyperbola defined by xy = 11/2 (i.e., y = +1/(2x)), with a single choice of sign (+) throughout. Additional linear transformations are then used to work out the geometry of other specific equations of the form Ax2 - By2 = +1 with A, B > 0. (a) Let R 1/V2 1/ V2 -1/V2 1/V2 ; the effect of TR : R2 -> R2 is clockwise rotation by 45 around the origin. For a point v = V1 U.2 , show that v lies on the hyperbola xy = 1/2 precisely when Rv lies on the curve H. defined by x2 - y2 = 1. Also show that v lies on the hyperbola xy = -1/2 precisely when Rv lies on the curve H_ defined by x2 - y? = -1. (Put together, these say H+ is exactly the output of applying the rotation TR to the graph of the function +1/(2x).)