Answer
455.1k+ views
Hint: - If three points are given as A,B,C then they will be collinear if the slopes of the line segment between 2 points are equal. I.e. slope of AB = slope of BC.
Given points are
${\text{A}}\left( {8,1} \right),{\text{ B}}\left( {3, - 4} \right),{\text{ and C}}\left( {2,k} \right)$
Now we know two points are collinear if their slopes are equal
Therefore slope of AB $ = $ Slope of BC
Collinearity of points: - Collinear points always lie on the same line.
Now we know
Slope between two points ${\text{ = }}\left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)$
Consider $A\left( {8,1} \right) \equiv \left( {{x_1},{y_1}} \right),{\text{ }}B\left( {3, - 4} \right) \equiv \left( {{x_2},{y_2}} \right),{\text{ }}C\left( {2,k} \right) \equiv \left( {{x_3},{y_3}} \right)$
Therefore slope of AB${\text{ = }}\left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right){\text{ = }}\dfrac{{ - 4 - 1}}{{3 - 8}} = \dfrac{{ - 5}}{{ - 5}} = 1$
Therefore slope of BC ${\text{ = }}\left( {\dfrac{{{y_3} - {y_2}}}{{{x_3} - {x_2}}}} \right){\text{ = }}\dfrac{{k - \left( { - 4} \right)}}{{2 - 3}} = \dfrac{{k + 4}}{{ - 1}} = - k - 4$
Points are collinear
Therefore slope of AB $ = $ Slope of BC
$\begin{gathered}
\Rightarrow 1 = - k - 4 \\
\Rightarrow k = - 1 - 4 = - 5 \\
\end{gathered} $
So, $k = - 5$ is the required answer.
Note: - If three points are collinear then the area formed by these points should be zero because collinear points always lie on the same line so the area formed by these points is zero. We can also use this property to find the collinearity of the points and the condition is $\left| {\begin{array}{*{20}{c}}
{{x_1}}&{{y_1}}&1 \\
{{x_2}}&{{y_2}}&1 \\
{{x_3}}&{{y_3}}&1
\end{array}} \right| = 0$, with the help of this we can easily calculate the collinearity of the given points.
Given points are
${\text{A}}\left( {8,1} \right),{\text{ B}}\left( {3, - 4} \right),{\text{ and C}}\left( {2,k} \right)$
Now we know two points are collinear if their slopes are equal
Therefore slope of AB $ = $ Slope of BC
Collinearity of points: - Collinear points always lie on the same line.
Now we know
Slope between two points ${\text{ = }}\left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)$
Consider $A\left( {8,1} \right) \equiv \left( {{x_1},{y_1}} \right),{\text{ }}B\left( {3, - 4} \right) \equiv \left( {{x_2},{y_2}} \right),{\text{ }}C\left( {2,k} \right) \equiv \left( {{x_3},{y_3}} \right)$
Therefore slope of AB${\text{ = }}\left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right){\text{ = }}\dfrac{{ - 4 - 1}}{{3 - 8}} = \dfrac{{ - 5}}{{ - 5}} = 1$
Therefore slope of BC ${\text{ = }}\left( {\dfrac{{{y_3} - {y_2}}}{{{x_3} - {x_2}}}} \right){\text{ = }}\dfrac{{k - \left( { - 4} \right)}}{{2 - 3}} = \dfrac{{k + 4}}{{ - 1}} = - k - 4$
Points are collinear
Therefore slope of AB $ = $ Slope of BC
$\begin{gathered}
\Rightarrow 1 = - k - 4 \\
\Rightarrow k = - 1 - 4 = - 5 \\
\end{gathered} $
So, $k = - 5$ is the required answer.
Note: - If three points are collinear then the area formed by these points should be zero because collinear points always lie on the same line so the area formed by these points is zero. We can also use this property to find the collinearity of the points and the condition is $\left| {\begin{array}{*{20}{c}}
{{x_1}}&{{y_1}}&1 \\
{{x_2}}&{{y_2}}&1 \\
{{x_3}}&{{y_3}}&1
\end{array}} \right| = 0$, with the help of this we can easily calculate the collinearity of the given points.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)