# If three points $\left( {k,2k} \right),\left( {2k,3k} \right),\left( {3,1} \right)$ are collinear, then k is equal to:

$

{\text{A}}{\text{. }} - 2 \\

{\text{B}}{\text{. }}1 \\

{\text{C}}{\text{. }}\dfrac{1}{2} \\

{\text{D}}{\text{. }} - \dfrac{1}{2} \\

$

Last updated date: 20th Mar 2023

•

Total views: 308.7k

•

Views today: 4.87k

Answer

Verified

308.7k+ views

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.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE