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Question
Find the equation of the straight line passing through the origin and bisecting the portion of the line ax + by + c = 0 intercepted between the coordinate axes.
Solution
The equation of the line passing through the origin is y = mx.
Let the line ax + by + c = 0 meet the coordinate axes at A and B.
So, the coordinates of A and B are \[A \left( - \frac{c}{a}, 0 \right) \text { and }B \left( 0, - \frac{c}{b} \right)\].
Now, the midpoint of AB is \[\left( - \frac{c}{2a}, - \frac{c}{2b} \right)\].
Clearly, \[\left( - \frac{c}{2a}, - \frac{c}{2b} \right)\] lies on the line y = mx.
\[\therefore - \frac{c}{2b} = m \times \frac{- c}{2a}\]
\[ \Rightarrow m = \frac{a}{b}\]
Hence, the equation of the required line is
\[y = \frac{a}{b}x\]
\[ \Rightarrow ax - by = 0\]
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