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Question
Find the coordinates of the point R on the line segment joining the points P(–1, 3) and Q(2, 5) such that PR = `3/5` PQ.
Solution
According to the question,
Given that,
PR = `3/5`PQ
⇒ `("PQ")/("PR") = 5/3`
⇒ `("PR" + "RQ")/("PR") = 5/3`
⇒ `1 + ("RQ")/("PR") = 5/3`
⇒ `("PQ")/("PR") = 5/3 - 1 = 2/3`
∴ RQ : PR = 2 : 3
or PR : RQ = 3 : 2
Suppose, R(x, y) be the point which divides the line segment joining the points P(–1, 3) and Q(2, 5) in the ratio 3 : 2
∴ (x, y) = `{(3(2) + 2(-1))/(3 + 2), (3(5) + 2(3))/(3 + 2)}` ...`[∵ "By internal section formula", {(m_2x_1 + m_1x_2)/(m_1 + m_2), (m_2y_1 + m_1y_2)/(m_1 + m_2)}]`
= `((6 - 2)/5, (15 + 6)/5)`
= `(4/5, 21/5)`
Hence, the required coordinates of the point R is `(4/5, 21/5)`.
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