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Question

If the point of intersection of the line r=(i+2j+3k)+λ(2i+j+2k) and the plane r(2i6j+3k)+5=0 lies on the plane r(i+75j+60k)α=0, then 19α+17 is equal to

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Solution

Given eq
given eq of line
r=(^i+2^j+3^k)+λ(2^i+^j+2^k)
comparing with eq of line we get
position vector A(1,2,3) a=2,b=1,c=2
eq become
x12=y21=z32=λ
passing the given eq to pointa^i+b^j+c^k we get
a=2λ+1,b=λ+2,c=2λ+3
passing given points to eq of planes
2a6b+3c+5=0
putting values of a,b,c
2(2λ+1)6(λ+2)+3(2λ+3)+5=0
4λ+26λ12+6λ+9+5=0
4λ=4
λ=1
a=2λ+1=2+1=1
b=λ+2=1+2=1
c=2λ+3=2+3=1
eq of planes
rn=an
comapring with given eq r(^i+75^j+60^k)=α
n=^i+75^j+60^k
an=α
a=^i+^j+^k
(^i+^j+^k)(^i+75^j+60^k)=α
α=1+75+60
α=134
19α+17=19×134+17=2546+17=2563

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