Question

In what ratio does the line $x-5y+15=0$ divide the join of $A(2,1)$ and $B(-3,6)?$ Also, find the co-ordinates of their point of intersection.

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 $D(x,y)$ be the point of intersection of the line $x-5y+15=0$

Let P divides the segment $A(2,1)$ and $B(-3,6)$ in the ratio $k:1$

Then by the formula of segment intersections, we have,

$x=\frac{-3k+2}{k+1}$ and $y=\frac{6k+1}{k+1}$

Substituting the values of x and y in the equation

$x-5y+15=0$, we have,

$\Rightarrow \frac{-3k+2}{k+1}-5\left(\frac{6k+1}{k+1}\right)+15=0$

$\Rightarrow \frac{-3k+2}{k+1}-5\left(\frac{6k+1}{k+1}\right)+\frac{15(k+1)}{k+1}=0$

$\Rightarrow -3k+2-5(6k+1)+15(k+1)=0$

$\Rightarrow 3k+2-30k-5+15k+15=0$

$\Rightarrow -12k+12=0$

$k=1$

Thus the points of intersection, $P(\frac{-3k+2}{k+1},\frac{6k+1}{k+1})=P(\frac{-1}{2},\frac{7}{2})$.