Question

Find the relation between x and y if the points \[\left( {x,y} \right)\] , \[\left( {1,2} \right)\] , and \[\left( {7,0} \right)\] are collinear?

Answer 1

Hint: Here in the equation, we have to find the relation between the variable x and y using the given points. By given data the points are collinear when the points are collinear The slope of any two points must be the same. By using the slope-point formula \[\left( {{y_2} - {y_1}} \right) = m\left( {{x_2} - {x_1}} \right)\] on simplification we get the required solution.

Complete step-by-step answer:
Collinear points are the points that lie on the same line. If two or more than two points lie on a line close to or far from each other, then they are said to be collinear.
If Three or more points are collinear, the slope of any two pairs of points is the same. With three points A, B and C, three pairs of points can be formed, they are: AB, BC and AC. If Slope of AB = slope of BC = slope of AC, then A, B and C are collinear points.
Consider the slope point formula:
 \[\left( {{y_2} - {y_1}} \right) = m\left( {{x_2} - {x_1}} \right)\] -----(1)
 \[y\] = y coordinate of second point
 \[{y_1}\] = y coordinate of point one
 \[m\] = slope
 \[x\] = x coordinate of second point
 \[{x_1}\] = x coordinate of point one
On rearranging equation (1) for slope m, then
 \[ \Rightarrow m = \dfrac{{\left( {{y_2} - {y_1}} \right)}}{{\left( {{x_2} - {x_1}} \right)}}\]
Consider the points \[A\left( {x,y} \right)\] , \[B\left( {1,2} \right)\] , and \[C\left( {7,0} \right)\] .
The slope of the line AB is
 \[ \Rightarrow m = \dfrac{{\left( {2 - y} \right)}}{{\left( {1 - x} \right)}}\] ------(3)
The slope of the line BC is
 \[ \Rightarrow m = \dfrac{{\left( {0 - 2} \right)}}{{\left( {7 - 1} \right)}}\] ------(4)
Given the points A, B and C are collinear then the slope of AB and BC are same then
By equation (3) and (4)
 \[ \Rightarrow \dfrac{{\left( {2 - y} \right)}}{{\left( {1 - x} \right)}} = \dfrac{{\left( {0 - 2} \right)}}{{\left( {7 - 1} \right)}}\]
 \[ \Rightarrow \dfrac{{\left( {2 - y} \right)}}{{\left( {1 - x} \right)}} = \dfrac{{ - 2}}{6}\]
On cross multiplication
 \[ \Rightarrow 6\left( {2 - y} \right) = - 2\left( {1 - x} \right)\]
 \[ \Rightarrow 12 - 6y = - 2 + 2x\]
On rearranging the equation
 \[ \Rightarrow 12 + 2 = 2x + 6y\]
 \[ \Rightarrow 14 = 2x + 6y\]
Divide both side by 2, we get
 \[ \Rightarrow 7 = x + 3y\]
Or
 \[ \Rightarrow x + 3y = 7\]
Hence, the relation between x and y when given points are collinear is \[x + 3y = 7\] .
So, the correct answer is “ \[x + 3y = 7\]”.

Note: Here we use the formula \[\left( {{y_2} - {y_1}} \right) = m\left( {{x_2} - {x_1}} \right)\] where \[{y_2},{y_1},{x_2}\,and\,{x_1}\] are the points where it will have the value of x and y. By substituting the values of the x and y we determine the equation for the given points. While simplifying we use the simple mathematical operations.