Question

If the points $\left( {{a_1},{b_1}} \right)$, $\left( {{a_2},{b_2}} \right)$ and $\left[ {\left( {{a_1} + {a_2}} \right),\left( {{b_1} + {b_2}} \right)} \right]$ collinear, show that ${a_1}{b_2} = {a_2}{b_1}$.

Answer 1

Hint: Here we have to obtain the given relation between the coordinates of the given points. As the given points are collinear, so the slopes of the lines joining these points are equal to each other. So we will first find the slope of the line joining the first two points and then the slope of the line joining the last two points and then we will equate both the slopes. Then we will simplify the equation obtained to get the required relation.

Formula used:
If $\left( {a,b} \right)$ and $\left( {c,d} \right)$ are two different points then the slope of the line joining these points will be equal to $\dfrac{{d - b}}{{c - a}}$.

Complete step by step solution:
Here we have to obtain the given relation between the coordinates of the given points.
The given three points are $\left( {{a_1},{b_1}} \right)$, $\left( {{a_2},{b_2}} \right)$ and $\left[ {\left( {{a_1} + {a_2}} \right),\left( {{b_1} + {b_2}} \right)} \right]$.
As the given points are collinear, so the slopes of the lines joining these points are equal to each other.
We know that if $\left( {a,b} \right)$ and $\left( {c,d} \right)$ are two different points then the slope of the line joining these points will be equal to $\dfrac{{d - b}}{{c - a}}$
We will find the slope of the line joining the first two points i.e. $\left( {{a_1},{b_1}} \right)$ and $\left( {{a_2},{b_2}} \right)$.
Therefore, the slope joining the points $\left( {{a_1},{b_1}} \right)$ and $\left( {{a_2},{b_2}} \right)$ $ = \dfrac{{{b_2} - {b_1}}}{{{a_2} - {a_1}}}$ …………….. $\left( 1 \right)$
Now, we will find the slope of the line joining the first two points i.e. $\left( {{a_2},{b_2}} \right)$ and $\left[ {\left( {{a_1} + {a_2}} \right),\left( {{b_1} + {b_2}} \right)} \right]$
Therefore, the slope joining the points $\left( {{a_2},{b_2}} \right)$ and $\left[ {\left( {{a_1} + {a_2}} \right),\left( {{b_1} + {b_2}} \right)} \right]$ $ = \dfrac{{\left( {{b_1} + {b_2}} \right) - {b_2}}}{{\left( {{a_1} + {a_2}} \right) - {a_2}}}$
On further simplification, we get
The slope joining the points $\left( {{a_2},{b_2}} \right)$ and $\left[ {\left( {{a_1} + {a_2}} \right),\left( {{b_1} + {b_2}} \right)} \right]$ $ = \dfrac{{{b_1}}}{{{a_1}}}$ …………… $\left( 2 \right)$
Now, we will equate equation 1 and equation 2 as the slopes of these equations are equal.
$ \Rightarrow \dfrac{{{b_2} - {b_1}}}{{{a_2} - {a_1}}} = \dfrac{{{b_1}}}{{{a_1}}}$
On cross multiplying the terms, we get
$ \Rightarrow {a_1}\left( {{b_2} - {b_1}} \right) = \left( {{a_2} - {a_1}} \right){b_1}$
Now, we will use the distributive property of multiplication to multiply the terms, we get
$ \Rightarrow {a_1} \cdot {b_2} - {a_1} \cdot {b_1} = {a_2} \cdot {b_1} - {a_1} \cdot {b_1}$
On further simplification, we get
$ \Rightarrow {a_1}{b_2} = {a_2}{b_1}$
Hence, we have proved the given relation between the coordinates of the points.

Note: Here we have obtained the slopes of the line joining the given points. Here the slope is defined as the number that represents the steepness and the direction of the line. If the value of the slope is greater than zero then the line goes up from left to right and if the value of the slope is greater than zero then that line always goes down from left to right i.e. line is decreasing.