Question

Find the ratio in which the line segment joining of the points $(1,2)$ and $(-2,3)$ is divided by the line $3x+4y=7$.

Answer 1

Let us consider that the point $A(x,y)$ divides the line segment joining the points $(1,2)$ and $(-2,3)$ in the ratio $k:1$

So the coordinates of point $(x,y)$ by section formula $\left(\frac{-2k+1}{k+1},\frac{3k+2}{k+1}\right)$

Since the point $A$ also lie on the line $3x+4y=7$

Therefore,

$3\left(\frac{-2k+1}{k+1}\right)+4\left(\frac{3k+2}{k+1}\right)=7$

$3\left(\frac{-2k+1}{k+1}\right)+4\left(\frac{3k+2}{k+1}\right)-7=0$

$3\left(-2k+1\right)+4\left(3k+2\right)-7\left(k+1\right)=0$

$-6k+3+12k+8-7k-7=0$

$-k+4=0$

$k=4$

Therefore, the line segment joining the points $(1,2),(-2,3)$ is divided by the by the line $3x+4y=7$ internally in the ratio of $4:1$.