Question

Show that the points are collinear \[\left( { - 5,1} \right)\left( {5,5} \right)\left( {10,7} \right)\].

Answer 1

Hint:
Here we will use the basic condition of the collinearity. Collinear points are the points which are lying on the same straight line. We will put the values of the coordinates in the equation of the condition of the collinearity to check whether the points are collinear or not.

Complete step by step solution:
Given points are \[\left( { - 5,1} \right)\left( {5,5} \right)\left( {10,7} \right)\].
We know that for the three points to be collinear they have to satisfy the condition of the collinearity.
The condition of the collinearity is \[{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right) = 0\] for the points \[\left( {{x_1},{y_1}} \right)\left( {{x_2},{y_2}} \right)\left( {{x_3},{y_3}} \right)\].
Now, we will put the value of the coordinates of the given points in this equation for the condition of the collinearity. Therefore, we get
\[ - 5\left( {5 - 7} \right) + 5\left( {7 - 1} \right) + 10\left( {1 - 5} \right) = 0\]
Simplifying the expression, we get
\[ \Rightarrow - 5\left( { - 2} \right) + 5\left( 6 \right) + 10\left( { - 4} \right) = 0\]
\[ \Rightarrow 10 + 30 - 40 = 0\]
Adding the terms, we get
\[ \Rightarrow 0 = 0\]
As the left hand side of the equation satisfies the right hand side of the equation, therefore, the given three points satisfies the condition of the collinearity.

Hence, the points \[\left( { - 5,1} \right)\left( {5,5} \right)\left( {10,7} \right)\] are collinear.

Note:
When the points do not lie on the same straight line then those points are known as non-collinear points. The points are said to be collinear if the slope of any two pairs of points is the same.
For example let three points A,B and C be the collinear points, then the slope of the line AB must be equal to the slope of the line BC must be equal to the slope of the line AC.
We need to keep in mind that the coordinates of the points is represented in the form of \[\left( {x,y} \right)\] where \[x\] is the intercept of the point on the X-axis and \[y\] is the intercept of the point on the Y-axis. We should note that always the \[x\] coordinate is written first in the coordinate representation.