Question

Find the ratio in which the point $(\frac{3}{4},\frac{5}{12})$divides the line segments joining the points $(\frac{1}{2},\frac{3}{2})$ 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

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

Let the required ratio be $m:n$ here, $x_1=\frac{1}{2},y_1=\frac{3}{2},x_1=2,x_2=-5$

So,

$\frac{m{x}_2+n{x}_1}{m+n}=\frac{3}{4}$

$\Rightarrow \frac{m\left(2\right)+n\left(\frac{1}{2}\right)}{m+n}=\frac{3}{4}$

$\Rightarrow 2m+\frac{n}{2}=\frac{3}{4}(m+n)$

$\Rightarrow \frac{4m+n}{2}=\frac{3m+3n}{4}$

$\Rightarrow 4m+n=\frac{3m+3n}{2}$

$\Rightarrow 8m+2n=3m+3n$

$\Rightarrow 5m=n$

$\Rightarrow \frac{m}{n}=\frac{1}{5}\Rightarrow m:n=1:5$