Question

Find the relation between $x$ and $y$ if the points $A(x,y),B(-5,7)$ and $C(-4,5)$ are collinear.

Answer 1

Area of triangle $=\frac{1}{2}\left[x_1\right(y_2-y_1)+x_2(y_3-y_1)+x_3(y_1-y_3\left)\right]$

$=\frac{1}{2}\left[x\right(7-5)+5(5-y)-4(4-7\left)\right]$

Given $A,B$ and $C$ are collinear, then area of triangle must be zero.

$\therefore \frac{1}{2}\left[x\right(7-5)+5(5-y)-4(4-7\left)\right]=0$

$\Rightarrow \frac{1}{2}[2x-25+5y-4y+25]=0$

$\Rightarrow \frac{1}{2}(2x+y+3)=0$

Then, relation between $x$ and $y$ is $2x+y+3=0$