(a) Find the coordinates of the point which divides the line joining (7, -5) and (-2, 7) externally in the ration 3 : 2.
(b) Without using calculators or mathematical tables, evaluate (frac{2}{1 + sqrt{2}}) – (frac{2}{2 + sqrt{2}}), leaving the answer in the form p + q(sqrt{n}), where p, q and n are integers.
Explanation
(a) If we let P(x, y) be the coordinates of the point, then we would have;
P(x, y) = ((frac{(3)(-2) – (2)(7)}{3 -2}), (frac{(3)(7) – (2)(5)}{3- 2})) = ((frac{-6 – 14}{1})), (frac{21 – 18}{1})) = (-20, 31)
(b), They took the L.C.M. and arrived at (frac{2}{1 – sqrt{2}}) – (frac{2}{2 – sqrt{2}}) = (frac{2(2 + sqrt{2}) – 2 (1 – sqrt{2})}{(1 – sqrt{2})(2 + sqrt{2})})
Simplify and rationalize to get -4 – (sqrt{2})