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Question
Find the ratio in which the point `P(3/4, 5/12)` divides the line segment joining the points `A(1/2, 3/2)` and B(2, –5).
Solution
Let `"P"(3/4, 5/12)` divide AB internally in the ratio m : n
Using the section formula, we get
`(3/4, 5/12) = ((2m - n/2)/(m + n), (-5m + 3/2n)/(m + n))` ...`[∵ "Internal section formula, the coordinates of point P divides the line segment joining the point" (x_1, y_1) "and" (x_2, y_2) "in the ratio" m_1 : m_2 "internally is" ((m_2x_1 + m_1x_2)/(m_1 + m_2), (m_2y_1 + m_1y_2)/(m_1 + m_2))]`
On equating, we get
`3/4 = (2m - n/2)/(m + n)` and `5/12 = (-5m + 3/2n)/(m + n)`
⇒ `3/4 = (4m - n)/(2(m + n))` and `5/12 = (-10m + 3n)/(2(m + n))`
⇒ `3/2 = (4m - n)/(m + n)` and `5/6 = (-10m + 3n)/(m + n)`
⇒ 3m + 3n = 8m – 2n and 5m + 5n = – 60m + 18n
⇒ 5n – 5m = 0 and 65m – 13n = 0
⇒ n = m and 13(5m – n) = 0
⇒ n = m and 5m – n = 0
Since, m = n does not satisfy.
∴ 5m – n = 0
⇒ 5m = n
∴ `"m"/"n" = 1/5`
Hence, the required ratio is 1 : 5.
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