Question

Find the ratio in which the line joining points $(a+b,b+a)$ & $(a-b,b-a)$ is divided by $(a,b)$

Answer 1

If a point $\text{P}(x,y)$ divides the line segment joining points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ then,

Using the section formula, we have $(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

$\Rightarrow (a,b)=(\frac{m(a-b)+n(a+b)}{m+n},\frac{m(b-a)+n(b+a)}{m+n})$

Equating, we get

$\Rightarrow a=\frac{m(a-b)+n(a+b)}{m+n}$

$\Rightarrow ma+na=ma-mb+na+nb$

$\Rightarrow 0=-mb+nb$

$\therefore mb=nb$ or $\frac{m}{n}=1$

Hence the ratio is $1:1$