Prove that the line $5 x - 2 y - 1 = 0$ is mid-parallel to the lines $5 x - 2 y - 9 = 0$ and $5 x - 2 y + 7 = 0$ .

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Solution

Let lines be $l_{1} = 5 x - 2 y - 1 = 0$
$l_{2} = 5 x - 2 y - 9 = 0$
$l_{3} = 5 x - 2 y + 7 = 0$
Slope of $l_{1} = 5 / 2$
Slope of $l_{2} = 5 / 2$
Slope of $l_{3} = 5 / 2$
dist. b/w $l_{1} & l_{2} = \frac{1 - 9 + 11}{\sqrt{25 + 4}} = \frac{8}{\sqrt{29}}$
dist b/w $l_{1} & l_{3} = \frac{1 - 1 - 71}{\sqrt{25 + 4}} = \frac{8}{\sqrt{29}}$
Hence $l_{1}$ is mid parallel to $l_{2} & l_{3}$ .

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