Question

If the X-axis divide the line segment joining the points $(2,-3)$ and $(5,6)$ in ration $p:q$, where $p$ and $q$ are co-primes, then find the value of $p+q$?

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

Substituting $(x_1,y_1)=(2,-3)$ and $(x_2,y_2)=(5,6)$ in the section formula, we get the point $(\frac{m\left(5\right)+n\left(2\right)}{m+n},\frac{m\left(6\right)+n(-3)}{m+n})=(\frac{5m+2n}{m+n},\frac{6m-3n}{m+n})$

As the point lies on x - axis, so the y -coordinate $=0$.

$=>\frac{6m-3n}{m+n}=0$

$=>6m=3n$ or $m:n=1:2$