Question

Find the ratio in which the point $P$ $\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points $A$ $\left(\frac{1}{2},\frac{3}{2}\right)$ and B $(2,-5)$.

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Therefore,

$\frac{3}{4}=\frac{m\times 2+n\times \frac{1}{2}}{m+n}$
$3m+3n=8m+4\times \frac{1}{2}n$
$3m+3n=8m+2n$
$3n-2n=8m-3m$
$n=5m$
$\frac{1}{5}=\frac{m}{n}$
$m:n=1:5$