Question

Determine the ratio in which y – x + 2 = 0 divides the line joining (3, –1) and (8, 9).​

Answer 1

$\underline{\underline{\sf{\color{magenta}{GIVEN :-}}}}$

y – x + 2 = 0 divides the line joining (3, –1) and (8, 9).​

$\underline{\underline{\sf{\color{magenta}{TO\ FIND :-}}}}$

The ratio in which y – x + 2 = 0 divides the line joining (3, –1) and (8, 9).​

$\underline{\underline{\sf{\color{magenta}{SOLUTION :-}}}}$

Let the line y – x + 2 = 0 divides the line segment joining A(3, –1) and B(8, 9)  in the ratio λ : 1 at a point P, then the co-ordinates of the point P are :

$\sf{\bigg(\dfrac{8\lambda+3}{\lambda+1} ,\dfrac{9\lambda-1}{\lambda+1} \bigg)}$

But P lies on y – x + 2 = 0. Therefore,

$\sf{\bigg(\dfrac{9\lambda-1}{\lambda+1}\bigg)- \bigg(\dfrac{8\lambda+3}{\lambda+1} \bigg)+2=0}$

$\sf{\longmapsto 9\lambda-1-8\lambda-3+2\lambda+2=0}$

$\sf{\longmapsto 3\lambda-2=0}$

$\sf{\longmapsto \lambda=\dfrac{2}{3} }$

Therefore, the required ratio is 2/3 : 1 (internally) since here λ is positive.

Answer 2

Let the required ratio be k:1

Then, the point of division is (8k+3/k+1, 9k-1/k+1)

This point must lie on y-x+2=0.

Therefore 9k-1/k+1 - 8k+3/k+1 +2-0 or k = 2/3.

So, the required ratio is 2/3:1

i.e. 2:3.