Question

If the coordinates of the points A, B, C, D, be \[\left( {a,\;b} \right),\;\left( {a',\;b'} \right),\;\left( { - a,\;b} \right)\;and\;\left( {a',\; - b'} \right)\] respectively, then the equation of the line bisecting the line segments AB and CD is
A) \[2a'y - 2bx = ab - a'b'\;\;\]
B) \[\;2ay - 2b'\;x = ab - a'b'\]
C) \[2ay - 2b'x = a'b - ab'\;\;\;\]
D) None of these

Answer 1

Hint: First calculate the midpoints of line segment AB and CD through which required line passes. Midpoint is a point which divides the line in equal proportion. Bisection means the point divides the line in two equal parts. Use coordinates of midpoint in order to find the slope of required line.

Formula used:Coordinates of midpoints is
Mid point = \[\left( {\dfrac{{\left( {{x_1} + {x_2}} \right)}}{2},\;\dfrac{{\left( {{y_1} + {y_2}} \right)}}{2}} \right)\] .
 where \[\left( {{x_1},{y_{1\;}}} \right),\;\left( {{x_2},{y_2}} \right)\]are coordinates of two points on line
 equation of straight line is:
\[y - {y_1} = m\left( {x - {x_1}} \right)\]
where ,
\[\left( {{x_1},\;{y_1}} \right)\]is coordinates through which line is passes
m is the slope of line

Complete step by step solution :
Given: coordinates of four points
Midpoint of AB is \[\left( {\dfrac{{\left( {{x_1} + {x_2}} \right)}}{2},\;\dfrac{{\left( {{y_1} + {y_2}} \right)}}{2}} \right)\]
Midpoint of AB = \[\left( {\dfrac{{\left( {a + a'} \right)}}{2},\;\dfrac{{\left( {b + b'} \right)}}{2}} \right)\]
Midpoint of CD =\[\left( {\dfrac{{\left( {{x_1} + {x_2}} \right)}}{2},\;\dfrac{{\left( {{y_1} + {y_2}} \right)}}{2}} \right)\]
Midpoint of CD=\[\left( {\dfrac{{\left( {a' - a} \right)}}{2},\;\dfrac{{\left( {b - b'} \right)}}{2}} \right)\]
According to question required line is passes through points having coordinates:
\[\left( {\dfrac{{\left( {a + a'} \right)}}{2},\;\dfrac{{\left( {b + b'} \right)}}{2}} \right)\] , \[\left( {\dfrac{{\left( {a' - a} \right)}}{2},\;\dfrac{{\left( {b - b'} \right)}}{2}} \right)\]
 Slope of required line = \[\dfrac{{\left( {{y_2} - {y_1}} \right)}}{{\left( {{x_2} - {x_1}} \right)}}\]
Equation of required line is :
\[y - {y_1} = m\left( {x - {x_1}} \right)\]
\[y - {y_1} = \left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)\left( {x - {x_1}} \right)\]
\[y - \dfrac{{\left( {b + b'\;} \right)}}{2} = \left( {\dfrac{{b - b' - b - b'}}{{a' - a - a - a'}}} \right)\left( {x - \dfrac{{a + a'}}{2}} \right)\]
On simplification we get
\[2ay - 2b'x - = ab - a'b'\]
Equation of required line :
\[2ay - 2b'x - = ab - a'b'\]



Thus, Option (B) is correct.

Note: Do not use the equation of line in any other form because it will become very difficult to find the equation of lines and sometimes one may not find the equation of line by using the other equation of lines. Slope of line is found using the coordinates of two points through which lines pass. Use the two point formula of a straight line because two points through which line passes are given indirectly in the question.