If the points P(-2, 1), Q(α, β) and R(4, -1) are collinear and α - β = -3, then the value of (α + β) is: 

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AAI Junior Assistant (Fire Service) Official Paper (Held On: 15 Nov 2022 Shift 1)
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  1. 2
  2. -1
  3. 1
  4. 0

Answer (Detailed Solution Below)

Option 2 : -1
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Detailed Solution

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Given:

Points: P(-2, 1), Q(α, β), R(4, -1)

α - β = -3

Formula used:

For collinear points, slope of PQ = slope of QR

Slope formula: (y2 - y1) / (x2 - x1)

Calculation:

Slope of PQ = (β - 1) / (α + 2)

Slope of QR = (-1 - β) / (4 - α)

Since P, Q, R are collinear:

⇒ (β - 1) / (α + 2) = (-1 - β) / (4 - α)

Cross-multiplying:

⇒ (β - 1)(4 - α) = (-1 - β)(α + 2)

⇒ 4β - β α - 4 + α = -α -2 - β α - 2β

⇒ 4β + α + α + 2β = -2 + 4

⇒ 3β + α = 1

Given that α - β = -3:

From α - β = -3 ⇒ α = β - 3

Substitute in 3β + α = 1:

⇒ 3β + (β - 3) = 1

⇒ 4β - 3 = 1

⇒ β = 1

So, α = 1 - 3 = -2

α + β = -2 + 1 = -1

∴ The correct answer is -1.

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