Question

Find the ratios in which the following straight lines divide the line segments joining the given points. State whether the points lie on the same side or on either side of the straight line.
$3x-4y=7;$ $(2,-7)$ and $(-1,3)$.

Answer 1

Equation of the line is $3x-4y=7$

Given points are $A(2,-7)$ and $B(-1,3)$.

The ratio in which the line $L=0$ divides the line join of point $A$ and $B$ is $-\frac{L_{11}}{L_{22}}=$

$-\frac{(a{x}_1+b{y}_1+c)}{(a{x}_2+b{y}_2+c)}$

$=-\frac{[3.2-4(-7)-7]}{[3-(-1)-4.3-7]}=-\frac{(6+28-27)}{-3-12-7}$

$=\frac{-27}{-22}=\frac{27}{22}$

The ratio is positive, therefore given points lie on opposite sides of the line.