Question

What is the equation of line whose midpoint is \[\left( {{x_1},{y_1}} \right)\] in between the axes?
A. \[\dfrac{x}{{{x_1}}} + \dfrac{y}{{{y_1}}} = 2\]
B. \[\dfrac{x}{{{x_1}}} + \dfrac{y}{{{y_1}}} = \dfrac{1}{2}\]
C. \[\dfrac{x}{{{x_1}}} + \dfrac{y}{{{y_1}}} = 1\]
D. None of these

Answer 1

Hint In the given question the midpoint of a straight line is given. We will find the x-intercept and y-intercept of the straight line by applying the midpoint formula. Then by using the equation of a line in intercept form, we will find the equation of a line.

Formula used
Midpoint Formula: The midpoint between the two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is: \[m = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\]
Equation of a line in intercept form is: \[\dfrac{x}{a} + \dfrac{y}{b} = 1\]

Complete step by step solution:
Given: The midpoint of a line between the axes is \[\left( {{x_1},{y_1}} \right)\].
Let \[P\left( {a,0} \right)\] be the x-axis intercept of the line and \[Q\left( {0,b} \right)\] be the y-axis intercept of the line.
Apply midpoint formula and calculate the coordinates of the x-intercept and y-intercept of the line \[PQ\].
\[\left( {{x_1},{y_1}} \right) = \left( {\dfrac{{a + 0}}{2},\dfrac{{0 + b}}{2}} \right)\]
\[ \Rightarrow \]\[\left( {{x_1},{y_1}} \right) = \left( {\dfrac{a}{2},\dfrac{b}{2}} \right)\]
\[ \Rightarrow \]\[a = 2{x_1}\] and \[b = 2{y_1}\]
Since, \[a = 2{x_1}\] is the x-intercept and \[b = 2{y_1}\] is the y-intercept.
So, apply the intercept form of an equation of a line.
\[\dfrac{x}{{2{x_1}}} + \dfrac{y}{{2{y_1}}} = 1\]
Simplify the above equation.
Multiply both sides by 2.
\[ \Rightarrow \]\[\dfrac{x}{{{x_1}}} + \dfrac{y}{{{y_1}}} = 2\]

Hence the correct option is option A.

Note: Students are often confused with the formula \[m = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\] and \[m = \left( {\dfrac{{{x_1} - {x_2}}}{2},\dfrac{{{y_1} - {y_2}}}{2}} \right)\] . But the correct midpoint formula is \[m = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\].