Question

Determine the ratio in which the point $\text{P}(a,-2)$ divides the line segment joining of points $\text{A}(-4,3)$ and $\text{B}(2,-4)$. Also find the value of $a$.

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

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

Given the points $A(-4,3)andB(2,-4)$

let $P(a,-2)$ divides the join of $A(-4,3)\equiv (x_1,y_1)$and $B(2,8)\equiv (x_2,y_2)$ in the ratio $K:1$

$\Rightarrow P(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$$=P(\frac{2K-4}{K+1},\frac{-4K+3}{K+1})$

$\therefore \frac{2K-4}{K+1}=a,\frac{-4K+3}{K+1}=-2$

$-4K+3=-2(K+1)$

$\Rightarrow -4K+3=-2K-2$

$\Rightarrow -2K=-5$

$\therefore K=\frac{5}{2}$

Hence the ration of divison is $5:2$

$\therefore a=\frac{2K-4}{K+1}=\frac{2\times \frac{5}{2}-4}{\frac{5}{2}+1}=\frac{(5-4)2}{5+2}=\frac{2}{7}$

$\therefore$ value of $a=\frac{2}{7}$