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Find the value of \( m \) if the points \( (5,1),(-2,-3) \) and \( (8,2 m) \) are collinear.
Given:
Points \( (5,1),(-2,-3) \) and \( (8,2 m) \) are collinear.
To do:
We have to find the value of $m$.
Solution:
Given points are: $( -2,\ -3),\ ( 5,\ 1),\ ( 8,\ 2m)$.
Here $x_1=-2,\ y_1=-3,\ x_2=5,\ y_2=1,\ x_3=8,\ y_3=2m$
If given points are collinear, then area of the triangle formed by the given points is zero.
$\Rightarrow \frac{1}{2}[x_1( y_2-y_3)+x_2( y_3-y_1)+x_3( y_1-y_2)]=0$
$\Rightarrow \frac{1}{2}[-2( 1-2m)+5( 2m-(-3))+8(-3-1)]=0$
$\Rightarrow \frac{1}{2}[-2+4m+5( 2m+3)+8(-4)]=0$
$\Rightarrow (-2+4m+10m+15-32)=0$
$\Rightarrow (14m-19)=0$
$\Rightarrow 14m=19$
$\Rightarrow m=\frac{19}{14}$
Therefore, the value of $m$ is $\frac{19}{14}$.
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