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Question
Find the equation to the straight line which bisects the distance between the points (a, b), (a', b') and also bisects the distance between the points (−a, b) and (a', −b').
Solution
Let the given points be A (a, b), B (a', b'), C (−a, b) and D (a', −b').
Let P and Q be the midpoints of AB and CD, respectively.
\[\therefore P \equiv \left( \frac{a + a^{\prime}}{2}, \frac{b + b^{\prime}}{2} \right)\]
\[Q \equiv \left( \frac{a^{\prime} - a}{2}, \frac{b - b^{\prime}}{2} \right)\]
The equation of the line passing through P and Q is
\[y - \frac{b + b^{\prime}}{2} = \frac{\frac{b - b'}{2} - \frac{b + b'}{2}}{\frac{a' - a}{2} - \frac{a' + a}{2}}\left( x - \frac{a + a^{\prime}}{2} \right)\]
\[ \Rightarrow 2y - b - b^{\prime} = \frac{b^{\prime}}{a}\left( 2x - a - a^{\prime} \right)\]
\[ \Rightarrow 2ay - 2 b^{\prime} x = ab - a^{\prime} b^{\prime}\]
Hence, the equation of the required straight line is \[2ay - 2 b^{\prime} x = ab - a^{\prime} b^{\prime}\]
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