Family of Planes Passing through the Intersection of Two Planes
The vector eq...
Question
The vector equation of the plane passing through the intersection of the planes →r⋅(^i+^j+^k)=1 and →r⋅(^i−2^j)=−2, and the point (1,0,2) is
A
→r⋅(^i−7^j+3^k)=73
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B
→r⋅(^i+7^j+3^k)=7
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C
→r⋅(3^i+7^j+3^k)=7
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D
→r⋅(^i+7^j+3^k)=73
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Solution
The correct option is B→r⋅(^i+7^j+3^k)=7 Family of planes passing through intersection of the given planes is {→r⋅(^i+^j+^k)−1}+λ{→r⋅(^i−2^j)+2}=0
The above curve passes through ^i+2^k. (3−1)+λ(1+2)=0 ⇒λ=−23
Hence, equation of the plane is 3{→r⋅(^i+^j+^k)−1}−2{→r⋅(^i−2^j)+2}=0 ⇒→r⋅(^i+7^j+3^k)=7
TRICK: Only option →r⋅(^i+7^j+3^k)=7 satisfies the point (1,0,2)