Question

Which of the following is true for the points X and Y if the coordinates of the midpoint P of \[\overline {XY} \] are \[\left( { - 2,3} \right)\]?
A) \[X\left( { - 4, - 2} \right)\] and \[Y\left( {0,4} \right)\]
B) \[X\left( { - 4,3} \right)\] and \[Y\left( {2,2} \right)\]
C) \[X\left( { - 6,2} \right)\] and \[Y\left( {2,4} \right)\]
D) \[X\left( {0,2} \right)\] and \[Y\left( { - 2,4} \right)\]

Answer 1

Hint: Here in this, we have to check the correct coordinates of X and Y point which have a midpoint P \[\left( { - 2,3} \right)\]. This can be solved by using a section formula \[P\left( {x,y} \right) = \left( {\dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}},\dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}}} \right)\] where \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] are the two point which the line segment join and, \[{m_1}:{m_2}\] are the ratios that point \[P\left( {x,y} \right)\] divides the line segment internally, the midpoint of line segments divides the line at ratio \[1:1\] respectively on substituting the values in formula and by simplification we get the required solution.

Complete step by step solution:
The coordinates of the point \[P\left( {x,y} \right)\] which divides the line segment joining the points \[A\left( {{x_1},{y_1}} \right)\]and \[B\left( {{x_2},{y_2}} \right)\] internally, in the ratio \[{m_1}:{m_2}\] are
\[P\left( {x,y} \right) = \left( {\dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}},\dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}}} \right)\] This is known as the section formula.
Now, we check the points X and Y if the coordinates of the midpoints P of \[\overline {XY} \] are \[\left( { - 2,3} \right)\]. The midpoint of the line segment divides the line at ratio \[{m_1}:{m_2} = 1:1\].
Let us consider points \[X\left( { - 4, - 2} \right)\] and \[Y\left( {0,4} \right)\], then by section formula
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{\left( 1 \right)\left( 0 \right) + \left( 1 \right)\left( { - 4} \right)}}{{1 + 1}},\dfrac{{\left( 1 \right)\left( 4 \right) + \left( 1 \right)\left( { - 2} \right)}}{{1 + 1}}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{1 - 4}}{2},\dfrac{{4 - 2}}{2}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( { - \dfrac{3}{2},\dfrac{2}{2}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( { - \dfrac{3}{2},1} \right)\]
Which is not true

Now, consider points \[X\left( { - 4,3} \right)\] and \[Y\left( {2,2} \right)\], then by section formula
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{\left( 1 \right)\left( 2 \right) + \left( 1 \right)\left( { - 4} \right)}}{{1 + 1}},\dfrac{{\left( 1 \right)\left( 2 \right) + \left( 1 \right)\left( 3 \right)}}{{1 + 1}}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{2 - 4}}{2},\dfrac{{2 + 3}}{2}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( { - \dfrac{2}{2},\dfrac{5}{2}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( { - 1,\dfrac{5}{2}} \right)\]
Which is not true.

Now, consider points \[X\left( { - 6,2} \right)\] and \[Y\left( {2,4} \right)\], then by section formula
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{\left( 1 \right)\left( 2 \right) + \left( 1 \right)\left( { - 6} \right)}}{{1 + 1}},\dfrac{{\left( 1 \right)\left( 4 \right) + \left( 1 \right)\left( 2 \right)}}{{1 + 1}}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{2 - 6}}{2},\dfrac{{4 + 2}}{2}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( { - \dfrac{4}{2},\dfrac{6}{2}} \right)\]
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( { - 2,3} \right)\]
Which is true.

Hence option (C) is correct.
Therefore, the midpoint of line of point \[X\left( { - 6,2} \right)\] and \[Y\left( {2,4} \right)\] is \[\left( { - 2,3} \right)\].

Note:
The section formula used to find any middle point of line segment. Remember if the midpoint of a line segment divides the line segment in the ratio \[{m_1}:{m_2} = 1:1\]. Therefore, the coordinates of the midpoint P of the join of the points \[A\left( {{x_1},{y_1}} \right)\] and \[B\left( {{x_2},{y_2}} \right)\] is
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{1 \cdot {x_2} + 1 \cdot {x_1}}}{{1 + 1}},\dfrac{{1 \cdot {y_2} + 1 \cdot {y_1}}}{{1 + 1}}} \right)\]
On simplification, we get
\[ \Rightarrow \,\,P\left( {x,y} \right) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\]
This can also be used when the middle point is located at the midpoint of the line segment.