The normal to the rectangular hyperbola xy = 8 at the point (4, 2) meets the asymp-
totes at M and N. Find the length of MN.

Equation of assymptotes to the rectangular hyperbola xy = 8 are x = 0 & y = 0.
Differentiate xy = 8 wrt x    x dy/dx + y.1 = 0.      dy/dx = –y/x

  Slope of tangent at (4,2) is –2/4 = –1/2

  Slope of normal at (4.2) is –1/(–1/2) = 2.

Equation of normal is y – 2 = 2(x – 4)    2x – y – 6 = 0.

It intersects x-axis at M(3,0) and y-axis at N(0,–6).

By distance formula, MN =   =   = 3