Show that the path of a moving point such that its distances from two lines $3 x - 2 y = 5$ and $3 x + 2 y = 5$ are equal is a straight line.

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Solution

Let P(h, k) be a moving point such that it is equidistant from the lines $3 x - 2 y - 5 = 0$ then

$∣ ∣ ∣ \frac{3 h - 2 k - 5}{\sqrt{9 + 4}} ∣ ∣ ∣ = ∣ ∣ ∣ \frac{3 h + 2 k - 5}{\sqrt{9 + 4}} ∣ ∣ ∣$

$| 3 h - 2 k - 5 | = | 3 h + 2 k - 5 |$

$4 k = 0$ or $6 h - 10 = 0$

$k = 0$

$3 h = 5$

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

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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.

Let P(h, k) be a moving point such that it is equidistant from the lines 3x−2y−5=0 then ∣∣∣3h−2k−5√9+4∣∣∣=∣∣∣3h+2k−5√9+4∣∣∣ |3h−2k−5|=|3h+2k−5| 4k=0 or 6h−10=0 k=0 3h=5 Hence, the path of the moving points are y=0 or 3x=5, which are straight lines.

Show that the path of a moving point such that its distances from two lines $3 x - 2 y = 5$ and $3 x + 2 y = 5$ are equal is a straight line.