Question

The points \[\left( { - a, - b} \right)\], \[\left( {0,0} \right)\], \[\left( {a,b} \right)\] and \[\left( {{a^2},ab} \right)\] are
A.Collinear
B.Vertices of a rectangle
C.Vertices of a parallelogram
D.None of these

Answer 1

Hint: First we will assume that the point be \[{\text{A}}\left( { - a, - b} \right)\], \[{\text{O}}\left( {0,0} \right)\], \[{\text{B}}\left( {a,b} \right)\] and \[{\text{C}}\left( {{a^2},ab} \right)\]. Then we will use the formula of slope of a line passing through points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is \[\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\] and then find the slope of lines joining OA, OB, and OC to find the required answer.

Complete step by step answer:

Let us assume that the point be \[{\text{A}}\left( { - a, - b} \right)\], \[{\text{O}}\left( {0,0} \right)\], \[{\text{B}}\left( {a,b} \right)\] and \[{\text{C}}\left( {{a^2},ab} \right)\].

We know the formula of slope of a line passing through points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is \[\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\].

First, we will compute the slope of the line joining OA by substituting the points in the above formula, we get

\[ \Rightarrow \dfrac{{ - b - 0}}{{ - a - 0}} = \dfrac{b}{a}\]

Finding the slope of the line joining OB by substituting the points in the above formula of the slope, we get

\[ \Rightarrow \dfrac{{b - 0}}{{a - 0}} = \dfrac{b}{a}\]

Substituting the points in the above formula of the slope to find the slope of the line joining OC by, we get

\[ \Rightarrow \dfrac{{ab - 0}}{{{a^2} - 0}} = \dfrac{b}{a}\]

Since the slope of the lines OA, OB, and OC are equal, they are all collinear.

Hence, option A is correct.

Note: In solving these types of questions, the key concept that we need to recall is that some pairs of points will always be collinear. So if all the slopes are equal then points are collinear. The student must remember that never equate the only slope without having a common point in all the slope calculations and say the slope is equal, which is wrong. So there must be a common point in all the calculations of the slope to check the collinearity of points.