Question
Download Solution PDFIf 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:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
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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