Question

Prove that the points $P\left( 2,-1 \right)$ , $Q\left( 1,-4 \right)$ and $R\left( 3,2 \right)$ are collinear.

Answer 1

Hint: We take two line segments $PQ$ and $QR$ and find out the slope of these two lines. If the slopes come out to be the same, we can declare the two line segments are part of the same line and thus, the three points are collinear.

Complete step-by-step answer:
The given points are $P\left( 2,-1 \right)$ , $Q\left( 1,-4 \right)$ and $R\left( 3,2 \right)$ . We have to prove that these three points are collinear. The collinear means that the points must lie on the same straight line. Now, there exists a line joining $P\left( 2,-1 \right)$ and $Q\left( 1,-4 \right)$ and another line joining $Q\left( 1,-4 \right)$ and $R\left( 3,2 \right)$ . These two lines, in general, are two different line segments which intersect at $Q\left( 1,-4 \right)$ . We can prove that the three points are collinear if we can prove that these two line segments are part of one and the same line.
Let us consider the line joining $P\left( 2,-1 \right)$ and $Q\left( 1,-4 \right)$ . We know that if we are provided with two points, then we can easily find the slope of the line joining the two points using the formula
$Slope=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Where, ${{y}_{2}},{{y}_{1}},{{x}_{2}},{{x}_{1}}$ are the coordinates of the two points. Let us assume $\left( {{x}_{1}},{{y}_{1}} \right)$ is $P\left( 2,-1 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $Q\left( 1,-4 \right)$ . The slope becomes,
$\Rightarrow Slope=\dfrac{-4-\left( -1 \right)}{1-2}$
$\Rightarrow Slope=\dfrac{-4+1}{-1}$
$\Rightarrow Slope=\dfrac{-3}{-1}$
$\Rightarrow Slope=3$
Let us consider the line joining $Q\left( 1,-4 \right)$ and $R\left( 3,2 \right)$ . Applying the same slope formula as above with $\left( {{x}_{1}},{{y}_{1}} \right)$ as $Q\left( 1,-4 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ as $R\left( 3,2 \right)$ , we get
$\begin{align}
  & \Rightarrow Slope=\dfrac{2-\left( -4 \right)}{3-1} \\
 & \Rightarrow Slope=\dfrac{2+4}{2} \\
 & \Rightarrow Slope=\dfrac{6}{2} \\
 & \Rightarrow Slope=3 \\
\end{align}$

Thus, we can see that the two distinct line segments have the same slope. Also, they have a common endpoint. Now, if two intersecting lines on the same plane have the same slope, then we can say that these two lines are nothing but the same line. Being the same line, we can say that the three points lie on the same line.
Therefore, we can conclude that the three given points are collinear.

Note: We must find out the slope of the two line segments using the two points in a sequential manner, otherwise this may lead to opposite sign slopes, and even sometimes different. We can also solve the problem by finding out the equation of the line passing through any two points. If the third point satisfies the equation, then we can say that the three points are collinear.