Question

Show that the points $\left( {a,0} \right)\;\left( {0,b} \right)\;\& \;\left( {3a - 2b} \right)$ are collinear. Also find the equation of the line containing them .

Answer 1

Hint: In this question use the concept of equation of lines i.e. find an equation passing through two points and then satisfy it with the third point as to satisfy the condition that all the three points are collinear.

Complete step-by-step solution -
According to the question , there are three points are given $\left( {a,0} \right)\;\left( {0,b} \right)\;\& \;\left( {3a - 2b} \right)$ and have to prove that they are collinear.
Hence , to prove all three points collinear we can do one thing , we will find the equation of lines passing through two points and then we will satisfy it with the third point.
Hence , Equation of line passing through $\left( {a,0} \right)\;\left( {0,b} \right)$.
We know the formula of Equation of line passing through points is $y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right)$
Let , $\left( {{x_1},{y_1}} \right) = \left( {a,0} \right)$
           $\left( {{x_2},{y_2}} \right) = \left( {0,b} \right)$
Now substituting the values in the formula, we get
$
   \Rightarrow y - 0 = \dfrac{{b - 0}}{{0 - a}}\left( {x - a} \right) \\
   \Rightarrow y = - \dfrac{b}{a}\left( {x - a} \right) \\
   \Rightarrow ay = - bx + ab \\
  \therefore ab = bx + ay \\
$
Hence the equation of line $ = $ $ ab = bx + ay$
Now we will see whether this equation of line satisfies the third point or not , if it satisfies the third point then all the points are collinear and that will be the equation of the line containing them .
Now checking third point i.e. $\left( {3a, - 2b} \right)$
Substituting the value in the equation of line we get
 $
   \Rightarrow b\left( {3a} \right) + a\left( { - 2b} \right) - ab = 0 \\
   \Rightarrow 3ab - 2ab - ab = 0 \\
   \Rightarrow 3ab - 3ab = 0 \\
 $
$ \Rightarrow 0 = 0$
Hence all the three points are collinear and the equation of line that is containing them is $ab = bx + ay$.

Note: It is advisable to know the basic concepts like collinear , equation of lines and straight line while involving into coordinate geometry questions as it helps in solving questions and saves time .