(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 - \(\sqrt{2}\)
