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

$|\frac{3 h - 2 k - 5}{\sqrt{3^{2} + 2^{2}}}| = |\frac{3 h + 2 k - 5}{\sqrt{3^{2} + 2^{2}}}| \Rightarrow |3 h - 2 k - 5| = |3 h + 2 k - 5| \Rightarrow 3 h - 2 k - 5 = \pm (3 h + 2 k - 5) \Rightarrow 3 h - 2 k - 5 = 3 h + 2 k - 5 and 3 h - 2 k - 5 = - (3 h + 2 k - 5) \Rightarrow k = 0 and 3 h = 5$

Hence, the path of the moving points are $3 x = 5 or y = 0$ These are straight lines.

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