Find the value of lambda if B, M and D are collinear

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Given a triangle OAB with OA = 5a and OB = 10b. D is the midpoint of AO and C is a point on AB such that AC: CB = 4:1. M is a point on OC such that OM = λOC. Find the value of λ if B, M and D are collinear.


D is the midpoint of AO ⇒ OD = ½OA
OD = 5a/2

AC: CB = 4:1 ⇒ OC = (1 * OA + 4 * OB)/ (1 + 4)
⇒ OC = a + 8b

OM = λOC ⇒ OM = λ/(λ + 1) * (a + 8b)

DM = DO + OM
⇒ DM = -5a/2 + λ/(λ + 1) * (a + 8b)

MB = MO + OB
⇒ MB = - λ/(λ + 1) * (a + 8b) + 10b

B, M and D are collinear ⇒ DM = kMB

Hence
-5a/2 + λ/(λ + 1) * (a + 8b) = k * (- λ/(λ + 1) * (a + 8b) + 10b)

Multiply throughout by 2(λ + 1) to remove the fractions
-5a(λ + 1) + 2λ * (a + 8b) = k * (- 2λ * (a + 8b) + 20(λ + 1)b)

Look at the coefficient of a
-5(λ + 1) + 2λ = k * - 2λ
2kλ = 5 + 3λ

Look at the coefficient of b
16λ = k * (- 16λ + 20(λ + 1))
16λ = k * (4λ + 20)
k * (λ + 5) = 4λ

Thus k = (5 + 3λ)/2λ = 4λ/(λ + 5)
Cross-multiplying
(5 + 3λ)(λ + 5) = 8λ²
3λ² + 20λ + 25 = 8λ²
5λ² - 20λ – 25 = 0
λ² - 4λ – 5 = 0
(λ – 5)(λ + 1) = 0
λ = 5 (or λ = -1, which would mean M is on BD produced)