Exams Joint Entrance Examination Mathematics 3D Geometry Question 286 Question 286 of 521 Medium NUMERICAL
If the equation of the plane passing through the line of intersection of the planes 2 x − 7 y + 4 z − 3 = 0 2x-7y+4z-3=0 2 x − 7 y + 4 z − 3 = 0 , 3 x − 5 y + 4 z + 11 = 0 3x-5y+4z+11=0 3 x − 5 y + 4 z + 11 = 0 and the point ( − 2 , 1 , 3 ) (-2,1,3) ( − 2 , 1 , 3 ) is a x + b y + c z − 7 = 0 ax+by+cz-7=0 a x + b y + cz − 7 = 0 , then the value of 2 a + b + c − 7 2a+b+c-7 2 a + b + c − 7 is:
Answer: 4
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( 2 x − 7 y + 4 z − 3 ) + λ ( 3 x − 5 y + 4 z + 11 ) = 0 (2x-7y+4z-3)+\lambda(3x-5y+4z+11)=0 ( 2 x − 7 y + 4 z − 3 ) + λ ( 3 x − 5 y + 4 z + 11 ) = 0 ; through ( − 2 , 1 , 3 ) (-2,1,3) ( − 2 , 1 , 3 ) :
\[(-4-7+12-3)+\lambda(-6-5+12+11)=0 \implies -2+12\lambda=0 \implies \lambda=\frac16.\]
\[\text{Plane: }15x-47y+28z-7=0.\]
\[a=15,\ b=-47,\ c=28 \implies 2a+b+c-7=30-47+28-7=4.\] Answer the question to unlock