Question

If the point $(-1,-1)$ divides the line segment joining the points $(4,4)$ and $(7,7)$ in the ratio of $a:b$, where $a$ and $b$ are co-primes, then find the value of $b-a$?

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 $(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$
Let the ratio be $k:1$

Substituting $(x_1,y_1)=(4,4)$ and $(x_2,y_2)=(7,7)$ in the section formula, we get

$(\frac{k\left(7\right)+1\left(4\right)}{k+1},\frac{k\left(7\right)+1\left(4\right)}{k+1})=(-1,-1)$

$(\frac{7k+4}{k+1},\frac{7k+4}{k+1})=(-1,-1)$

Comparing the x - coordinate,

$\Rightarrow \frac{7k+4}{k+1}=-1$

$\Rightarrow 7k+4=-k-1$

$8k=-5$

$k=-\frac{5}{8}$

Hence, the ratio $a:b$ is $5:8$ externally

Therefore, the value of $b-a=3$