Question

Find the ratio in which the join of the points $ \left( {1,2} \right) $ and $ \left( { - 2,3} \right) $ is divided by the line $ 3x + 4y = 7 $

Answer 1

Hint: In the given question, we are required to find the ratio in which a line divides the join of two points whose coordinates are given to us. Firstly, we find the coordinates of a point at which the line divides the line segment joining the two given points. Then, we find out the ratio in which the join of the two points is divided. The required task can be done using the section formula where we can use a predefined formula given the coordinates of the end points of the line segment and the coordinates of the point at which the line divides the join of endpoints.

Complete step by step solution:
We are given the points $ \left( {1,2} \right) $ and $ \left( { - 2,3} \right) $ .
We find out the equation of the line passing through these two points.
We know the two point form of the line, where we can find the equation of a straight line given the coordinates of two points lying on it. The two point form of the line can be represented as: \[\left( {y - {y_1}} \right) = \dfrac{{\left( {{y_2} - {y_1}} \right)}}{{\left( {{x_2} - {x_1}} \right)}}\left( {x - {x_1}} \right)\] where $ \left( {{x_1},{y_1}} \right) $ and $ \left( {{x_2},{y_2}} \right) $ are the coordinates of the two points.
So, the equation of the line passing through $ \left( {1,2} \right) $ and $ \left( { - 2,3} \right) $ is \[\left( {y - 2} \right) = \dfrac{{\left( {3 - 2} \right)}}{{\left( { - 2 - 1} \right)}}\left( {x - 1} \right)\] .
Now, simplifying the equation of the straight line, we get,
 \[ \Rightarrow - 3\left( {y - 2} \right) = \left( {x - 1} \right)\]
Opening the brackets, we get,
 \[ \Rightarrow - 3y + 6 = x - 1\]
 \[ \Rightarrow x + 3y - 7 = 0\]
Now, the equation of the other line is $ 3x + 4y = 7 $ .
Now, we find the point of intersection of these two lines by solving the equations of both.
Substituting the value of x from equation \[x + 3y - 7 = 0\] into the equation $ 3x + 4y = 7 $ , we get,
 $ \Rightarrow 3\left( {7 - 3y} \right) + 4y = 7 $
 $ \Rightarrow 21 - 9y + 4y = 7 $
Isolating the terms consisting of y, we get,
 $ \Rightarrow 5y = 14 $
 $ \Rightarrow y = \dfrac{{14}}{5} $
Now, substituting the value of y in the equation $ 3x + 4y = 7 $ , we get,
 $ \Rightarrow 3x + 4\left( {\dfrac{{14}}{5}} \right) = 7 $
Opening the bracket, we get,
 $ \Rightarrow 3x + \dfrac{{56}}{5} = 7 $
 $ \Rightarrow 3x = 7 - \dfrac{{56}}{5} = \dfrac{{35 - 56}}{5} $
Simplifying the calculations, we get,
 $ \Rightarrow 3x = \dfrac{{ - 21}}{5} $
 $ \Rightarrow x = \dfrac{{ - 7}}{5} $
So, the coordinates of the point at which the line $ 3x + 4y = 7 $ divides the join of $ \left( {1,2} \right) $ and $ \left( { - 2,3} \right) $ are $ \left( { - \dfrac{7}{5},\dfrac{{14}}{5}} \right) $ .
Now, we have to find the ratio in which the join of the endpoints $ \left( {1,2} \right) $ and $ \left( { - 2,3} \right) $ is divided at the point $ \left( { - \dfrac{7}{5},\dfrac{{14}}{5}} \right) $ .
So, let the ratio in which the point $ \left( { - \dfrac{7}{5},\dfrac{{14}}{5}} \right) $ divides the line segment be $ k:1 $ .
Using section formula, Coordinates of point P \[ = \left( {\dfrac{{n{x_1} + m{x_2}}}{{m + n}},\dfrac{{n{y_1} + m{y_2}}}{{m + n}}} \right)\] , where $ \left( {{x_1},{y_1}} \right) $ and $ \left( {{x_2},{y_2}} \right) $ are the coordinates of the endpoints of the line segment and $ m:n $ is the ratio in which the Point P divides the line segment.
So, Coordinates of Point P \[ = \left( {\dfrac{{1\left( 1 \right) + k\left( { - 2} \right)}}{{k + 1}},\dfrac{{1\left( 2 \right) + k\left( 3 \right)}}{{k + 1}}} \right)\]
Opening the brackets and simplifying the calculations, we get,
 \[ = \left( {\dfrac{{1 - 2k}}{{k + 1}},\dfrac{{2 + 3k}}{{k + 1}}} \right)\]
Now, equating this with the actual coordinates of Point P found already, we get,
 \[\dfrac{{1 - 2k}}{{k + 1}} = \dfrac{{ - 7}}{5}\] and \[\dfrac{{2 + 3k}}{{k + 1}} = \dfrac{{14}}{5}\]
 \[ \Rightarrow 5 - 10k = - 7k - 7\] and \[ 10 + 15k = 14k + 14\]
 \[ \Rightarrow 3k = 12\] and \[k = 4\]
 \[ \Rightarrow k = 4\]
Hence, the value of k comes out to be \[4\] from both the equations.
Hence, the ratio in which the join of the points $ \left( {1,2} \right) $ and $ \left( { - 2,3} \right) $ is divided by the line $ 3x + 4y = 7 $ is $ 4:1 $ .
So, the correct answer is “ $ 4:1 $ ”.

Note: We must remember the section formula and its applications in order to solve the problem. We should understand the language of the question carefully as without understanding the problem,
one might not be able to solve it properly. We should know how to solve equations of two given straight lines by substitution method to tackle the problem.