Question

Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

Answer 1

Hint: To solve the given question, we will first find out what collinearity is or what collinear points are. Then we will make use of the fact that if three points are collinear then the slope of the line formed by any two out of three points will be the same. Thus, we will calculate the slope by taking two pairs of the points and we will equate them. To find the slope between the points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right),\] we will use the formula shown.
\[\text{Slope}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
After doing this, we will get a linear relation between x and y.

Complete step-by-step answer:
Before we start to solve the given question, we will first find out the meaning of the collinearity of three points. Collinearity of a set of points is the property of the points lying on a single line. Thus, we can say that three or more points will be collinear if they lie on a single straight line. Now, to get the relation between x and y, we will make use of the fact that if three points are collinear, then the slope of the line formed by any two out of three points will be the same. Thus, first, we will calculate the slope. We know that if, two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] are given, then the slope of the line formed by these points will be given by
\[\text{Slope}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
Let us assume that the slope of the line formed by the points (x, y) and (7, 0) is \[{{m}_{1}}.\] Thus, we will get,
\[{{m}_{1}}=\dfrac{y-0}{x-7}\]
\[\Rightarrow {{m}_{1}}=\dfrac{y}{x-7}.....\left( i \right)\]
Let us assume that the slope of the line formed by the points (x, y) and (1, 2) will be \[{{m}_{2}}.\] Thus, we will get,
\[{{m}_{2}}=\dfrac{y-2}{x-1}.....\left( ii \right)\]
Now, both the slopes are equal, i.e. \[{{m}_{1}}={{m}_{2}}.\] Thus, from (i) and (ii), we will get,
\[\Rightarrow \dfrac{y}{x-7}=\dfrac{y-2}{x-1}\]
On cross multiplying, we will get,
\[\Rightarrow y\left( x-1 \right)=\left( y-2 \right)\left( x-7 \right)\]
\[\Rightarrow yx-y=yx-7y-2x+14\]
\[\Rightarrow -y=-7y-2x+14\]
\[\Rightarrow 2x+6y=14\]
On dividing the above equation by 2, we will get,
\[\Rightarrow x+3y=7\]
Thus, we have got the desired relation between x and y.


Note: We can also approach the question in an alternate way as shown. If three points lie on a line, i.e. they are collinear, then they cannot form a triangle. Thus, the area of the triangle formed by collinear points will be zero. Thus, area = 0.
\[\dfrac{1}{2}\left| \begin{matrix}
   x & y & 1 \\
   7 & 0 & 1 \\
   1 & 2 & 1 \\
\end{matrix} \right|=0\]
\[\Rightarrow \left| \begin{matrix}
   x & y & 1 \\
   7 & 0 & 1 \\
   1 & 2 & 1 \\
\end{matrix} \right|=0\]
\[\Rightarrow x\left[ 0\times 1-1\times 2 \right]-y\left[ 7\times 1-1\times 1 \right]+1\left[ 7\times 2-1\times 0 \right]=0\]
\[\Rightarrow x\left( -2 \right)-y\left( 6 \right)+14=0\]
\[\Rightarrow 2x+6y=14\]
\[\Rightarrow x+3y=7\]