Family of Planes Passing through the Intersection of Two Planes
A plane P : a...
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:x−y+z−2=0 Equation of plane P passing through the intersection of planes P1 and P2 is P1+λP2=0 ⇒(x+y+z+3)+λ(x−y+z−2)=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 3x−y+3z=1 ∴a+b+c=3−1+3=5