3) The hyperbola xy=1 has asymptotes x=0 and y=0. Let M and N be points on the positive branch with the secant line MN intersecting the asymptotes at T and U. WLOG let the three disjoint segments of this secant in the first quadrant be TM, MN & NU with TM and NU lying outside the conic section. Book II proposition 8 gives that TM=NU. Let SNR be the tangent to the hyperbola at N where S is its intersection with the y-axis and R its intersection with the x-axis. Use a straightforward limit argument (and II-8) to show that SN=NR. Then show, without calculus, that the slope of the tangent line to y=1/x at (a,1/a) is -1/a ².

4) Solve geometrically the system x-y=7, xy=18 (Consider a rectangle ABCD with area 18, put a point E on BC and a point F on DA so that ABEF is a square and EC has length 7, BC=AD=x and AB=BE=EF=FA=CD=y). Hint: look at the quadratic formula as a guide.