Question

Find in what ratio the line 3x+y=9 divides the line joining the points (1,3) (2,7)

Answer 1

Answer :-

3 : 4

Solution :-

Let the ratio of which 3x + y = 9 divides the line (1,3) (2,7) be k : 1

Using Section formula

$\sf P(x,y) = \bigg( \dfrac{m x_2 + nx_1 }{m + n} ,\dfrac{m y_2 + ny_1 }{m + n} \bigg)$

(1,3) (2,7)

Here,

• m = k
• n = 1
• x2 = 2
• x1 = 1
• y2 = 7
• y1 = 3

Substituting the values

$\implies \sf P(x,y) = \bigg( \dfrac{k(2) +1(1) }{k + 1} ,\dfrac{k(7) + 1(3) }{k + 1} \bigg)$

$\implies \sf P(x,y) = \bigg( \dfrac{2k +1 }{k + 1} ,\dfrac{7k+ 3}{k + 1} \bigg)$

Comparing on both sides

$\implies \sf x = \dfrac{2k +1 }{k + 1} \qquad ,y = \dfrac{7k+ 3}{k + 1}$

Now, substituting the values of x- coordinates and y - coordination in 3x + y = 9

$\implies \sf 3 \bigg(\dfrac{2k +1 }{k + 1} \bigg) + \dfrac{7k+ 3}{k + 1} = 9$

$\implies \sf \dfrac{6k +3}{k + 1} + \dfrac{7k+ 3}{k + 1} = 9$

$\implies \sf \dfrac{6k +3 + 7k + 3}{k + 1} = 9$

$\implies \sf \dfrac{13k + 6}{k + 1} = 9$

$\implies \sf 13k + 6= 9k + 9$

$\implies \sf 13k - 9k = 9 - 6$

$\implies \sf 4k = 3$

$\implies \sf k = \dfrac{3}{4}$

Therefore the ratio which divides the line is 3/4 : 1 or 3 : 4.