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Question
Show that the path of a moving point such that its distances from two lines 3x − 2y = 5 and 3x + 2y = 5 are equal is a straight line.
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Solution
Let P(h, k) be the moving point such that it is equidistant from the lines 3x − 2y = 5 and 3x + 2y = 5
\[\left| \frac{3h - 2k - 5}{\sqrt{3^2 + 2^2}} \right| = \left| \frac{3h + 2k - 5}{\sqrt{3^2 + 2^2}} \right|\]
\[ \Rightarrow \left| 3h - 2k - 5 \right| = \left| 3h + 2k - 5 \right|\]
\[ \Rightarrow 3h - 2k - 5 = \pm \left( 3h + 2k - 5 \right)\]
\[ \Rightarrow 3h - 2k - 5 = 3h + 2k - 5 and 3h - 2k - 5 = - \left( 3h + 2k - 5 \right)\]
\[ \Rightarrow k = 0 \text{ and } 3h = 5\]
Hence, the path of the moving points are \[3x = 5 \text{ or } y = 0\] These are straight lines.
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