Question

Determine the ratio in which the point P(m, 6) divides the join of A(– 4, 3) and B(2, 8). Also find the value of m.

(a) 2 : 5 ; m = 3 

(b) 3 : 2 ; m = \(-\frac{2}{5}\)

(c) 1 : 2 ; m = \(\frac{1}{4}\)

(d) 2 : 1 ; m = \(\frac{2}{3}\)

Answer 1

(b) 3 : 2 ; m = \(-\frac{2}{5}\)

Let P(m, 6) divides AB in the ratio k : 1. 

Then co-ordinates of P are \(\bigg(\)\(\frac{2k-4}{k+1}\), \(\frac{8k+3}{k+1}\)\(\bigg)\)

Given, co-ordinates of P are (m, 6) ⇒\(\frac{2k-4}{k+1}\) = m     ...(i)

and \(\frac{8k+3}{k+1}\) = 6 ⇒ 8k + 3 = 6k + 6 ⇒ 2k = 3 ⇒ k = \(\frac{3}{2}\)

∴ Required ratio is \(\frac{3}{2}\) : 1 = 3 : 2

Now, \(\frac{2k-4}{k+1}\) = m ⇒ \(\frac{2\times\frac{3}{2}-4}{\frac{3}{2}+1}\) = \(\frac{3-4}{\frac{5}{2}}\) = \(\frac{-2}{5}\)

∴ m = \(\frac{-2}{5}\).