Answer

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Hint- Here, we will be using the formula for slope between two points.

Let the given three points be ${\text{P}}\left( {k,2k} \right),{\text{Q}}\left( {2k,3k} \right),{\text{R}}\left( {3,1} \right)$

As we know that the slope of any line joining two points ${\text{A}}\left( {a,b} \right)$ and ${\text{B}}\left( {c,d} \right)$ is given by $m = \dfrac{{d - b}}{{c - a}}$.

Slope of line PQ is ${m_{PQ}} = \dfrac{{3k - 2k}}{{2k - k}} = \dfrac{k}{k} = 1$

Slope of line QR is ${m_{QR}} = \dfrac{{1 - 3k}}{{3 - 2k}}$

Slope of line PR is ${m_{PR}} = \dfrac{{1 - 2k}}{{3 - k}}$

For all the three given points to be collinear, the slope of lines made by joining any two points will be equal.

$ \Rightarrow {m_{PQ}} = {m_{QR}} = {m_{PR}} \Rightarrow 1 = \dfrac{{1 - 3k}}{{3 - 2k}} = \dfrac{{1 - 2k}}{{3 - k}}$

When taking $ \Rightarrow 1 = \dfrac{{1 - 3k}}{{3 - 2k}} \Rightarrow 3 - 2k = 1 - 3k \Rightarrow k = - 2$

And when $ \Rightarrow 1 = \dfrac{{1 - 2k}}{{3 - k}} \Rightarrow 3 - k = 1 - 2k \Rightarrow k = - 2$

Therefore, the value of $k$ for which the given three points are collinear is $ - 2$.

Hence, option A is correct.

Note- In these types of problems where collinearity of points is the condition, the key feature is to make the slope of the lines joining any two points equal in order to find the unknown. In the above problem, while considered different slopes equality the value of the unknown comes out to be the same.

Let the given three points be ${\text{P}}\left( {k,2k} \right),{\text{Q}}\left( {2k,3k} \right),{\text{R}}\left( {3,1} \right)$

As we know that the slope of any line joining two points ${\text{A}}\left( {a,b} \right)$ and ${\text{B}}\left( {c,d} \right)$ is given by $m = \dfrac{{d - b}}{{c - a}}$.

Slope of line PQ is ${m_{PQ}} = \dfrac{{3k - 2k}}{{2k - k}} = \dfrac{k}{k} = 1$

Slope of line QR is ${m_{QR}} = \dfrac{{1 - 3k}}{{3 - 2k}}$

Slope of line PR is ${m_{PR}} = \dfrac{{1 - 2k}}{{3 - k}}$

For all the three given points to be collinear, the slope of lines made by joining any two points will be equal.

$ \Rightarrow {m_{PQ}} = {m_{QR}} = {m_{PR}} \Rightarrow 1 = \dfrac{{1 - 3k}}{{3 - 2k}} = \dfrac{{1 - 2k}}{{3 - k}}$

When taking $ \Rightarrow 1 = \dfrac{{1 - 3k}}{{3 - 2k}} \Rightarrow 3 - 2k = 1 - 3k \Rightarrow k = - 2$

And when $ \Rightarrow 1 = \dfrac{{1 - 2k}}{{3 - k}} \Rightarrow 3 - k = 1 - 2k \Rightarrow k = - 2$

Therefore, the value of $k$ for which the given three points are collinear is $ - 2$.

Hence, option A is correct.

Note- In these types of problems where collinearity of points is the condition, the key feature is to make the slope of the lines joining any two points equal in order to find the unknown. In the above problem, while considered different slopes equality the value of the unknown comes out to be the same.

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