Chapter: Co-ordinate Geometry· Mathematics Part 2 (Geometry)
Answer: (i) k = 5 (ii) 2 : 1
(i) Slope of AC \(= \dfrac{8 - (-1)}{7 - 1} = \dfrac{9}{6} = \dfrac{3}{2}\). Slope of AB \(= \dfrac{5 - (-1)}{k - 1} = \dfrac{6}{k - 1}\).
A, B, C are collinear, so slope of AB = slope of AC: \(\dfrac{6}{k - 1} = \dfrac{3}{2}\) \(\therefore 3(k - 1) = 12\) \(\therefore k - 1 = 4\) \(\therefore k = 5\). So B(5, 5).
(ii) Let B divide AC in the ratio m : n. Using x co-ordinates: \(5 = \dfrac{7m + 1 \cdot n}{m + n}\) \(\therefore 5m + 5n = 7m + n\) \(\therefore 4n = 2m\) \(\therefore m : n = 2 : 1\).
Check with y: \(\dfrac{2(8) + 1(-1)}{3} = \dfrac{15}{3} = 5\). ✓ So B divides AC in the ratio 2 : 1.