Question

If the points (k,2-2k), (1-k, 2k) and (-k-4, 6-2k) are collinear, possible values of k are ..............
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Answer 1

Points A(k,2-2k), B (1-k, 2k) and C(-k-4, 6-2k) are collinear.
Then, slope of line AB = slope of line BC
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$\Rightarrow \frac{2k-(2-2k)}{1-k-k}=\frac{6-2k-2k}{-k-4-1+k}$
$\Rightarrow \frac{4k-2}{1-2k}=\frac{6-4k}{-5}$
$\Rightarrow -20k+10=6-4k-12k+8k^2$
$\Rightarrow 8k^2+4k-4=0$
$\Rightarrow 2k^2+k-1=0$
$\Rightarrow 2k(k+1)-1(k+1)=0$
$\Rightarrow (2k-1)(k+1)=0$
$\Rightarrow 2k-1=0andk+1=0$
$\Rightarrow k=\frac{1}{2}andk=-1$
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We can also say, slope of AB = slope of AC

$\Rightarrow \frac{2k-(2-2k)}{1-k-k}=\frac{6-2k-(2-2k)}{-k-4-k}$
$\Rightarrow \frac{4k-2}{1-2k}=\frac{4}{-2k-4}$
$\Rightarrow -8k^2+4k-16k+8=4-8k$
$\Rightarrow 8k^2+4k-4=0$
$\Rightarrow 2k^2+k-1=0$
we got the same equation of previous case
So,

$k=\frac{1}{2}$ and $k=-1$ are the only possible values of $k$
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when given points are collinear.