Find the relation between $x$ and $y$ if the points $A( x,\ y),\ B( -5,\ 7)$ and $C( -4,\ 5)$ are collinear.
Given: Points $A( x,\ y),\ B( -5,\ 7)$ and $C( -4,\ 5)$ are collinear.
To do: To find the relation between $x$ and $y$.
Solution:
Area of triangle $=\frac{1}{2}[x_1( y_2-y_1)+x_2( y_3-y_1)+x_3( y_1-y_3)]$
$=\frac{1}{2}[x(7-5)+5(5-y)-4(4-7)$
Given A,B and C are collinear, then area of triangle must be zero.
$\therefore \frac{1}{2}[x(7-5)+5(5-y)-4(4-7)]=0$
$\Rightarrow \frac{1}{2}[2x-25+5y-4y+25]=0$
$\Rightarrow \frac{1}{2}(2x+y+3)=0$
Then, relation between $x$ and $y$ is $2x+y+3=0$
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