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Question
Prove that the lines 2x + 3y = 19 and 2x + 3y + 7 = 0 are equidistant from the line 2x + 3y= 6.
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Solution
Let \[d_1\] be the distance between lines 2x + 3y = 19 and 2x + 3y = 6,
while \[d_2\] is the distance between lines 2x + 3y + 7 = 0 and 2x + 3y = 6
\[ \Rightarrow d_1 = \left| \frac{- 13}{\sqrt{13}} \right| = \sqrt{13} \text{ and } d_2 = \left| \frac{13}{\sqrt{13}} \right| = \sqrt{13}\]
Hence, the lines 2x + 3y = 19 and 2x + 3y + 7 = 0 are equidistant from the line 2x + 3y = 6
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