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Question : 31 of 160
Marks:
+1 ,
-0
Solution:
Position vectors of
A , B , C and
D are
‌ a = 7 − 4 + 7 ‌ b = − 6 + 10 ‌ c = − − 3 + 4 ‌ d = 5 − + ‌ respectively. ‌ ∵ Let point of intersection of
A C and
B D be
P
∴ Position of vector of
P is
( ‌ ) or
‌ ‌ ‌ ∴ ‌ − λ − 3 λ + 4 λ + 7 − 4 + 7 λ + 1
‌ = ‌ ‌ ‌ = ‌ ‌ ⇒ − λ µ + 7 µ − λ + 7 = 5 λ µ + λ + 5 µ + 1 ‌ 6 λ µ + 2 λ − 2 µ = 6 ‌ 3 λ µ + λ − µ = 3 ‌ ‌ ‌ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ( i ) ‌ ‌ = ‌ ‌ ⇒ − 3 λ µ − 4 µ − 3 λ − 4 = − λ µ − 6 λ − µ − 6 ‌ 2 λ µ − 3 λ + 3 µ = 2 ‌ ‌ ‌ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ( i i ) ‌ ‌ = ‌ ‌ ⇒ 4 λ µ + 7 + 7 µ + 4 λ = λ µ + 10 λ + µ + 10 ‌ 3 λ µ − 6 λ + 6 µ = 3 ‌ λ µ − 2 λ + 2 µ = 1 ‌ ‌ ‌ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ( i i i ) ‌ λ ( µ − 2 ) = 1 − 2 µ ‌ λ = ‌ By Eq. (i)
‌ 3 µ ( ‌ ) + ‌ − µ = 3 ‌ ⇒ 3 µ − 6 µ 2 + 1 − 2 µ − µ 2 + 2 µ = 3 µ − 6 ‌ ⇒ µ = ± 1 ‌ ‌ If ‌ µ = 1 , ‌ ‌ λ = 1 ‌ and ‌ µ = − 1 , ‌ ‌ λ = − 1 ∴ Position vector of
P = ‌ ( − λ + 7 ) + ( − 3 λ − 4 ) + ( 4 λ + 7 ) λ + 1
∴ p + q + r = ‌ − λ + 7 − 3 λ − 4 + 4 λ + 7 λ + 1
= ‌ = 5 ( ‌ At ‌ λ = 1 ) © examsiri.com
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