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Question

Find the equation of the plane passing through the line of intersection of the planes and and parallel to x -axis.

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Solution

The given planes are :

r ( i ^ + j ^ + k ^ )=1 r ( i ^ + j ^ + k ^ )1=0 (1)

r ( 2 i ^ +3 j ^ k ^ )+4=0(2)

The equation of plane passing through the intersection of two planes ( A 1 x+ B 1 y+ C 1 z d 1 =0 ) and ( A 2 x+ B 2 y+ C 2 z d 2 =0 ) is given by,

( A 1 x+ B 1 y+ C 1 z d 1 )+λ( A 2 x+ B 2 y+ C 2 z d 2 )=0(3)

Substitute the values of equation (1) and equation (2) in equation (3).

r ( i ^ + j ^ + k ^ )1+λ[ r ( 2 i ^ +3 j ^ k ^ )+4 ]=0 r [ ( 2λ+1 ) i ^ +( 3λ+1 ) j ^ +( 1λ ) k ^ ]+( 4λ1 )=0 (4)

The direction ratios of the new plane are given by,

( 2λ+1 ),( 3λ+1 ),( 1λ )

The plane is parallel to the x axis. The normal of the plane is perpendicular to the x axis.

The direction ratios of x are 1, 0 and 0.

1( 2λ+1 )+0( 3λ+1 )+0( 1λ )=0 2λ+1=0 2λ=1 λ= 1 2 (5)

Substitute the value of λ from equation (5) to equation (4).

r [ ( 2( 1 2 )+1 ) i ^ +( 2( 1 2 )+1 ) j ^ +( 1( 1 2 ) ) k ^ ]+( 4( 1 2 )1 )=0 r [ ( 1+1 ) i ^ +( 3 2 +1 ) j ^ +( 1+ 1 2 ) k ^ ]+( 21 )=0 r [ 0 i ^ 1 2 j ^ + 3 2 k ^ ]3=0 r [ j ^ +3 k ^ ]6=0

Further simplify the above expression.

r [ j ^ 3 k ^ ]+6=0

The above equation of plane in Cartesian form is given by,

y3z+6=0

Thus, the equation of plane passing through the line of intersection of planes r ( i ^ + j ^ + k ^ )1=0 and r ( 2 i ^ +3 j ^ k ^ )+4=0 is y3z+6=0.


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