Question

Find the ratio in which \[P\left( {4,m} \right)\] divides the line segment joining the points \[A\left( {2,3} \right)\] and \[B\left( {6, - 3} \right)\]. Hence find ‘m'.

Answer 1

Hint: To solve this problem we will start with applying the section formula, by assuming the point \[P\left( {4,m} \right)\] is dividing the line segment joining \[A\left( {2,3} \right)\] and \[B\left( {6, - 3} \right)\] in the ratio \[k:1.\] Then, after getting the value of k, we will use to find the value of m.

Complete step-by-step answer:
We need to find the ratio in which \[P\left( {4,m} \right)\] divides the line segment joining the points \[A\left( {2,3} \right)\] and \[B\left( {6, - 3} \right)\], and also we need to find the value of ‘m'. We will use the section formula here, because here a point \[P\left( {x,y} \right)\] is dividing the line segment joining \[A\left( {{x_1},{y_1}} \right)\] and \[B\left( {{x_2},{y_2}} \right)\] in the ratio m:n.
The general formula of section formula is mentioned below.
\[\Rightarrow \left( {x,y} \right) = \left( {\dfrac{{m{x_1} + n{x_2}}}{{m + n}},\dfrac{{m{y_1} + n{y_2}}}{{m + n}}} \right)\]
Here, let the point \[P\left( {4,m} \right)\] is dividing the line segment joining \[A\left( {2,3} \right)\] and \[B\left( {6, - 3} \right)\] in the ratio \[k:1.\] So, on applying the values in the above formula, we get
\[\Rightarrow P\left( {4,m} \right) = \left( {\dfrac{{k(6) + 1(2)}}{{k + 1}},\dfrac{{k( - 3) + 1(3)}}{{k + 1}}} \right)\]
\[ \Rightarrow 4 = \dfrac{{6k + 2}}{{k + 1}}\]
\[
 \Rightarrow 4k + 4 = 6k + 2 \\
\Rightarrow 2k = 2 \\
\Rightarrow k = 1 \\
\]
So, the ratio in which \[P\left( {4,m} \right)\] divides the line segment joining the points \[A\left( {2,3} \right)\] and \[B\left( {6, - 3} \right)\]is \[1:1.\]
Now, after applying the value of k, we get
$\Rightarrow m = \dfrac{{ - 3(1) + 3(1)}}{{1 + 1}}$
$\Rightarrow m = 0$
So, the value of m is \[0.\]
Thus, the ratio in which \[P\left( {4,m} \right)\] divides the line segment joining the points \[A\left( {2,3} \right)\] and \[B\left( {6, - 3} \right)\]is \[1:1\] and the value of m is \[0.\]

Note: In the solutions, we have applied the section formula. Let us understand in detail.
The coordinates of the point P(x,y) which divides the line segment joining the points \[A\left( {{x_1},{y_1}} \right)\] and \[B\left( {{x_2},{y_2}} \right)\] in ratio of m:n are
\[\left( {\dfrac{{m{x_1} + n{x_2}}}{{m + n}},\dfrac{{m{y_1} + n{y_2}}}{{m + n}}} \right)\]$.$
This formula is called section formula. This formula tells us about the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m:n.