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Question

A plane P:ax+by+cz=1 passes through the intersection of planes r(^i+^j+^k)=3 and r(^i^j+^k)=2. If plane P divides the line segment joining M(3,0,2) and N(0,3,1) in the ratio 2:1 internally, then (a+b+c) is equal to

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Solution

P1:x+y+z+3=0
P2:xy+z2=0
Equation of plane P passing through the intersection of planes P1 and P2 is
P1+λP2=0
(x+y+z+3)+λ(xy+z2)=0 (1)

Let point R divides the line segment MN in 2:1 internally, then
R(2(0)+1(3)2+1,2(3)+1(0)2+1,2(1)+1(2)2+1)
R(1,2,0)


As point R lies on the plane P
6+λ(3)=0
λ=2

Hence, equation of plane P is 3xy+3z=1
a+b+c=31+3=5

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