Question

Question 11
Find the ratio in which the point $P(\frac{3}{4},\frac{5}{12})$
divides the line segment joining the points $A(\frac{1}{2},\frac{3}{2})$ and B(2, -5).

Answer 1

Let $P(\frac{3}{4},\frac{5}{12})$ divide AB internally in the ratio m:n.
Using the section formula, we get
$(\frac{3}{4},\frac{5}{12})=(\frac{2m-\frac{n}{2}}{m+n},\frac{-5m+\frac{3}{2}n}{m+n})$
[$\because$ By internal section formula, the coordinates of point P dividing the line segment joining the point $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m_1:m_2$ internally is
$(\frac{m_2{x}_1+m_1{x}_2}{m_1+m+2},\frac{m_2{y}_1+m_1{y}_2}{m_1+m+2})]\text{On equating, we get}$
$\frac{3}{4}=\frac{2m-\frac{n}{2}}{m+n}and\frac{5}{12}=\frac{-5m+\frac{3}{2}n}{m+n}\Rightarrow \frac{3}{4}=\frac{4m-n}{2(m+n)}and\frac{5}{12}=\frac{-10m+3n}{2(m+n)}\Rightarrow \frac{3}{2}=\frac{4m-n}{m+n}and\frac{5}{6}=\frac{-10m+3n}{m+n}\Rightarrow 3m+3n=8m-2nand5m+5n=-60m+18n\Rightarrow 5n-5m=0and65m-13n=0\Rightarrow n=mand13(5m-n)=0\Rightarrow \text{m =n does not satisfy.}\text{Since, m =n does not satisfy.}\therefore 5m-n=0\Rightarrow 5m=n\therefore \frac{m}{n}=\frac{1}{5}\text{Hence, the required ratio is 1:5.}$