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Question

The equation of the plane which passes through the line of intersection of the planes rn1=q1rn2=q2 and is parallel to the line of intersection of the line of intersection of the planes rn3=q2 and rn4=q4 given that [n1n3n4)(n2n3n4] is

A
rn2q2=rn3q3
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B
rn1q1=[n1n2n3](rn2q2)
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C
rn1q1=rn2q2
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D
rn3q3=rn1q1
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Solution

The correct option is D rn1q1=rn2q2
Equation of any plane through the intersection of rn1=q1,r2n2=q2 is of the form rn1+λrn2=q1+λq2(1)
where λ is a parameter.
So n1+λ.n2 is normal to the plane (1) Any plane containing planes rn3=q3rn4=q4 is of the form
r(n3+μn4)=q3+μq4 Hence we must have n1+λn2=k(n3+μn4) for some k
[n1+λ.n2][n,3×n4]=0
[n1,n3,n4]+λ[n2,n3,n4]=0
λ=1
Putting this value in (1), we have equation of required plane as
rn1q1=(rn2q2) or
[n2n3n4](rn1q1)=[n1n3n4](rn2q2)

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