Question

In what ratio is the line joining the points (2, 3) and (4, -5) divided by the line passing through the points (6, 8) and (-3, -2).

Answer 1

$\text{The equation of the line joining the points (6, 8) and (-3, -2) is}y-8=\frac{-2-8}{-3-6}(x-6)\Rightarrow 10x-9y+12=0\text{Let 10x - 9y + 12 = 0 divide the line joining the points (2, 3) and (4, -5) at point P in the ratio k : 1}\therefore P=(\frac{4k+2}{k+1},\frac{-5k+3}{k+1})\text{P lies on the line 10x - 9y + 12 = 0}\therefore 10\left(\frac{4k+2}{k+1}\right)-9\left(\frac{-5k+3}{k+1}\right)+12=0\Rightarrow 40k+20+45k-27+12k+12=0\Rightarrow 97k+5=0\Rightarrow k=-\frac{5}{97}\text{Hence, the line joining the points (2, 3) and (4, -5) is divided by the passing through the points (6, 8) and (-3, -2) in the ratio 5 : 97 externally.}$