In what ratio does the point $(-4, 6)$ divide the line segment joining the points $A (-6, 10)$ and $B (3, -8)$?

Given :

Point $(-4,6)$ divides the line segment joining the points A$(-6,10)$ and B$(3,-8)$.

To find :

We have to find the ratio of division.

Solution :

Let $(-4,6)$ divides AB in the ratio m:n. 

The section formula is,

$(x, y) = \frac{m x_{2} + n x_{1}}{m + n} , \frac{m y_{2} + n y_{1}}{m + n} $

Here,

$(x, y) = (-4,6)$ ; $A (x_{1}, y_{1}) = A(-6,10)$ ; $B(x_{2}, y_{2}) =  B(3,-8)$

$(-4, 6) = \frac{m (3) + n(-6)}{m + n} , \frac{m (-8) + n (10)}{m + n} $

On comparing,

$-4 = \frac{3m-6n}{m + n}$

$-4(m + n) = 3m-6n$

$-4m-4n = 3m-6n$

$4m+3m+4n-6n = 0$

$7m-2n = 0$

$7m = 2n$

$\frac{m}{n} = \frac{2}{7}$     

$m : n = 2 : 7$

The required ratio is 2:7.

Updated on: 10-Oct-2022

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