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Question
In what ratio does the line x - y - 2 = 0 divide the line segment joining the points A (3, 1) and B (8, 9)?
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Solution
Let the line x - y - 2 = 0 divide the line segment joining the points A (3, 1) and B (8, 9) in the ratio k : 1 at P.
Then, the coordinates of P are
`"p" ((8"k"+3)/("k"+1),(9"k"-1)/("k"+1))`
Since, P lies on the line x - y - 2 = 0 we have:
` ((8"k"+3)/("k"+1)) - ((9"k"-1)/("k"+1)) -2=0`
⇒ 8k + 3 - 9k + 1 - 2k - 2 = 0
⇒ 8k - 9k - 2k + 3 + 1 - 2 = 0
⇒ - 3k + 2 = 0
⇒ - 3k = - 2
`⇒ "k" = 2/3`
So, the required ratio is `2/3:1` which is equal to 2 : 3.
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