The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is
Question:

The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is

(a) −2 : 3

(b) −3 : 2

(c) 3 : 2

(d) 2 : 3

Solution:

The co-ordinates of a point which divided two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ internally in the ratio $m: n$ is given by the formula,

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

Here it is said that the point (4, 5) divides the points A(2,3) and B(7,8). Substituting these values in the above formula we have,

$(4,5)=\left(\left(\frac{m(7)+n(2)}{m+n}\right),\left(\frac{m(8)+n(3)}{m+n}\right)\right)$

Equating the individual components we have,

$4=\frac{m(7)+n(2)}{m+n}$

$4 m+4 n=7 m+2 n$

$3 m=2 n$

$\frac{m}{n}=\frac{2}{3}$

Hence the correct choice is option (d).