Let the line 2x + 3y – 30 = 0 divide the join of A(3, 4) and B(7, 8) at point C(p, q) in the ratio k : 1. Then,
p = \(\frac{7k+3}{k+1}\), q = \(\frac{8k+4}{k+1}\)
As the point C lies on the line 2x + 3y – 30 = 0, it satisfies the given equation, i.e,
2 x \(\bigg(\frac{7k+3}{k+1}\bigg)\) + 3\(\bigg(\frac{8k+4}{k+1}\bigg)\) - 30 = 0
⇒ 14k + 6 + 24k + 12 – 30k – 30 = 0
⇒ 8k – 12 = 0 ⇒ k = \(\frac{12}{8}\) = \(\frac{3}{2}\)
∴ The line 2x + 3y – 30 = 0 divides the line joining A(3, 4) and B(7, 8) in the ratio \(\frac{3}{2}\) : 1. i.e. 3 : 2 at C.
Now the co-ordinates of C are \(\bigg(\frac{7\times\frac{3}{2}+3}{\frac{3}{2}+1},\frac{8\times\frac{3}{2}+4}{\frac{3}{2}+1}\bigg)\) = \(\big(\frac{27}{5},\frac{32}{5}\big)\).
