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Question
M is the mid-point of the line segment joining the points A(–3, 7) and B(9, –1). Find the coordinates of point M. Further, if R(2, 2) divides the line segment joining M and the origin in the ratio p : q, find the ratio p : q.
Solution
Given, M is the mid-point of the line segment joining the points A(−3, 7) and B(9, −1).
The co-ordinates of point M are
`((-3 + 9)/2, (7 - 1)/2)`
= `(6/2, 6/2)`
= (3, 3)
Also, given that, R(2, 2) divides the line segment joining M and the origin in the ratio p : q.
∴ `(2, 2) = ((p xx 0 + q xx 3)/(p + q),(p xx 0 + q xx 3)/(p + q))`
`=> (p xx 0 + q xx 3)/(p + q) = 2`
`=> (3q)/(p + q) = 2`
`=>` 3q = 2p + 2q
`=>` 3q – 2q = 2p
`=>` q = 2p
`=> p/q = 1/2`
Thus the ratio p : q is 1 : 2.
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