Question

$P$ and $Q$ are the points with coordinates $\left( {2, - 1} \right)$ and $\left( { - 3,4} \right)$. Find the coordinates of the point $R$ such that $PR$ is $\dfrac{2}{5}$ of $PQ$.

Answer 1

Hint:
Here, we will solve the given relation between $PR$ and $PQ$ to find the ratio in which the point \[R\] divides the line segment $PQ$ internally. Then, we will use Section formula to find the required coordinate of point $R$. We will then substitute the given coordinates in the section formula and solve it further to find the required coordinates of the point $R$.

Formula Used:
Section Formula: Coordinates of point $R = \dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}$, where $R$ is a point dividing the line segment.

Complete step by step solution:
It is given a line segment $PQ$ such that the coordinates of Point $P = \left( {2, - 1} \right)$ and the coordinates of point $Q = \left( { - 3,4} \right)$.
Now, there is a point $R$ which divides the line segment $PQ$ such that $PR$ is $\dfrac{2}{{{5^{th}}}}$of $PQ$.
Hence, we can write this mathematically as:
$PR = \dfrac{2}{5} \times PQ$
But we know that $PQ = PR + RQ$. So,
$ \Rightarrow PR = \dfrac{2}{5} \times \left( {PR + RQ} \right)$
Simplifying the equation, we get
$ \Rightarrow 5PR = 2PR + 2RQ$
Subtracting the like terms, we get
$ \Rightarrow 3PR = 2RQ$
On cross multiplication, we get
$ \Rightarrow \dfrac{{PR}}{{RQ}} = \dfrac{2}{3}$
Therefore, we get the ratio, $PR:RQ = 2:3$

Now, substituting $\left( {{x_1},{y_1}} \right) = \left( {2, - 1} \right)$ , $\left( {{x_2},{y_2}} \right) = \left( { - 3,4} \right)$ and the ratio $m:n = 2:3$ in the formula Coordinates of Point $R = \dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}$, we get,
Coordinates of point $R = \dfrac{{\left( 2 \right)\left( { - 3} \right) + \left( 3 \right)\left( 2 \right)}}{{2 + 3}},\dfrac{{\left( 2 \right)\left( 4 \right) + 3\left( { - 1} \right)}}{{2 + 3}}$
Multiplying the terms, we get
$ \Rightarrow $ Coordinates of point $R = \dfrac{{ - 6 + 6}}{5},\dfrac{{8 - 3}}{5}$
Simplifying the expression, we get
$ \Rightarrow $ Coordinates of point $R = \left( {0,1} \right)$
Therefore, the required coordinates of point $R$ are $\left( {0,1} \right)$
Hence, this is the required answer.

Note:
In this question, three points are given with their respective coordinates. We should take care while solving the question, that we have to substitute the correct coordinates in the correct place. For example, in the section formula, if we substitute $\left( {{x_1},{y_1}} \right)$ in such a way that the $x$ coordinate is of point $P$ and the $y$ coordinate is of point $Q$, then, our answer will be wrong. Similarly, while writing the ratios, we should keep in mind that $2:3$ is not the same as $3:2$, thus, we should solve it carefully.