+4 marks−1 if incorrectNumericalPrevious-year question
Intersection Condition for Coplanar Lines in 3D and Optimization of a Product
If the lines (x-1)/2=(2-y)/(-3)=(z-3)/α and (x-4)/5=(y-1)/2=z/β intersect, then the magnitude of the minimum value of 8 α β is _______.
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Hint 1 of 4
What are the standard symmetric forms of the two lines?
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Correct answer
The lines intersect when 2α−3β=−6, yielding a minimum value of −18 for 8αβ, whose magnitude is 18.
Option analysis
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Solution
StepWorking
01Given
Two intersecting lines:
Line 1: 2x−1=−32−y=αz−3⟹2x−1=3y−2=αz−3, passing through a1=(1,2,3) with direction vector b1=⟨2,3,α⟩.
Line 2: 5x−4=2y−1=βz, passing through a2=(4,1,0) with direction vector b2=⟨5,2,β⟩.
02Goal
Find the magnitude of the minimum value of the expression E=8αβ.
03Approach
Since the two lines intersect, they must be coplanar. Hence, the scalar triple product [a2−a1,b1,b2]=0. Using this condition, we obtain a linear relation between α and β. Then, express 8αβ as a quadratic in β and find its minimum value.
04Execute
Vector joining the points on the lines is a2−a1=(4−1)i^+(1−2)j^+(0−3)k^=3i^−j^−3k^ (or −3i^+j^+3k^).
Set up the coplanarity determinant:
25−3321αβ3=0
Expanding along the rows:
2(6−β)−3(15+3β)+α(5+6)=012−2β−45−9β+11α=011α−11β−33=0⟹α−β=3⟹α=β+3
05Execute
Substitute α=β+3 into E=8αβ:
E=8(β+3)β=8(β2+3β)=8(β+23)2−8(49)=8(β+23)2−18
The minimum value of E occurs when β=−23, giving Emin=−18.
The magnitude of the minimum value is ∣Emin∣=∣−18∣=18.
✓Verify
Check critical point: dβd[8(β2+3β)]=8(2β+3)=0⟹β=−3/2. At β=−3/2, α=3/2, so 8αβ=8(3/2)(−3/2)=−18. Magnitude is ∣−18∣=18.
Hints that build this answer step by step
What are the standard symmetric forms of the two lines?
2x−1=3y−2=αz−3 and 5x−4=2y−1=βz−0
What condition on α and β guarantees that the two lines intersect?
2α−3β=−6
Given 2α=3β−6, what is the minimum value of 8αβ?
−18
What is the magnitude of the minimum value of 8αβ?
The standard symmetric form of a line requires the coefficient of y in the numerator to be +1. Multiplying numerator and denominator by −1 gives 3y−2.
