Question

Find the coordinates of the point which divides the join of \[\left( { - 1,7} \right)\] and \[\left( {4, - 3} \right)\] in the ratio \[2:3.\]

Answer 1

Hint: By directly, we use the formula the point which divides the line segment in the ratio is \[m:n.\]
The coordinate point in the middle of the line segment, it divides the line in the ratio in two equal parts, but the point which divides the lines in any desired ratio.

Complete step-by-step answer:
We know that, the coordinates of the point dividing the line segment joining the points \[{x_1},{y_1}\] and \[{x_2},{y_2}\] in the ratio \[{m_{1}} : {m_{2}}\] is given by \[\left( {\dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}},\dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}}} \right)\].
This formula is for the point which divides the line in any desired ratio.
          

Given \[({x_1},{y_1}){\text{ }} = {\text{ }}\left( { - 1,7} \right)\]
\[({x_2},{y_2}) = {\text{ }}\left( {4, - 3} \right)\]
\[({m_1},{m_2}) = {\text{ }}\left( {2,3} \right)\]
Coordinates of point of intersection of line segment is
\[\left( {\dfrac{{(2)(4) + (3)( - 1)}}{{2 + 3}},\dfrac{{(2)( - 3) + (3)(7)}}{{2 + 3}}} \right)\]
On some simplification we get the values,
=\[\left( {\dfrac{{8 + ( - 3)}}{5},\dfrac{{( - 6) + 21}}{5}} \right)\]
Adding the numerators we get,
=\[\left( {\dfrac{{8 - 3}}{5},\dfrac{{ - 6 + 21}}{5}} \right)\]
On some simplification we get,
=\[\left( {\dfrac{5}{5},\dfrac{{15}}{5}} \right)\]
Divide the terms we get,
=\[(1,3)\]
Hence, the coordinate of the point \[\left( {1,3} \right)\] which divides the join of \[( - 1,7)\] and \[(4, - 3)\] in the ratio is \[2:3\].

Note: The formula we use in the sum is the section formula, that is the coordinates of the point \[P(x,y)\] which divides the line segment joining the points \[A({x_1},{y_1})\] and \[B({x_2},{y_2})\] in the ratio \[{m_1}:{m_2}\] are known as the section formula.
Each point on a number line is assigned a number. In the same way, each point in a plane is assigned a pair of numbers.
The coordinates for the origin are $\left( {0,0} \right)$.
Suppose, the midpoint of the line segment is the point that divides a line segment in two equal halves. But the section formula, it locates the point dividing the line segment in any desired ratio.