Can we find a line normal to the curves y = cosh x and y = sinh x?
The gradient of y = cosh x at x = a is sinh a, so the equation of the normal is sinh a (y - cosh a) + (x - a) = 0. Similarly, the normal to y = sinh x at x = b is cosh b (y - sinh b) + (x - b) = 0. For these two equations to be the same (so that the normals coincide) we require: sinh a = cosh b and a + sinh a cosh a = b + sinh b cosh b, or b - a = sinh a cosh a - sinh b cosh b = cosh a cosh b - sinh a sinh b = cosh (b - a). But that is impossible since cosh x > x for all x.
40th Putnam 1979
© John Scholes
4 Dec 1999