Exams Joint Entrance Examination Mathematics 3D Geometry Question 257 Question 257 of 521 Medium MCQ
Let the plane P : r ⃗ ⋅ a ⃗ = d P:\ \vec r\cdot\vec a=d P : r ⋅ a = d contain the line of intersection of the two planes r ⃗ ⋅ ( i ^ + 3 j ^ − k ^ ) = 6 \vec r\cdot(\hat i+3\hat j-\hat k)=6 r ⋅ ( i ^ + 3 j ^ − k ^ ) = 6 and r ⃗ ⋅ ( − 6 i ^ + 5 j ^ − k ^ ) = 7 \vec r\cdot(-6\hat i+5\hat j-\hat k)=7 r ⋅ ( − 6 i ^ + 5 j ^ − k ^ ) = 7 . If the plane P P P passes through the point ( 2 , 3 , 1 2 ) \left(2,3,\dfrac12\right) ( 2 , 3 , 2 1 ) , then the value of 13 ∣ a ⃗ ∣ 2 d \dfrac{13|\vec a|^2}{d} d 13∣ a ∣ 2 is equal to:
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Family: x ( 1 − 6 λ ) + y ( 3 + 5 λ ) + z ( − 1 − λ ) − ( 6 + 7 λ ) = 0 x(1-6\lambda)+y(3+5\lambda)+z(-1-\lambda)-(6+7\lambda)=0 x ( 1 − 6 λ ) + y ( 3 + 5 λ ) + z ( − 1 − λ ) − ( 6 + 7 λ ) = 0 . Through ( 2 , 3 , 1 2 ) \left(2,3,\tfrac12\right) ( 2 , 3 , 2 1 ) :
\[2(1-6\lambda)+3(3+5\lambda)+\frac12(-1-\lambda)-(6+7\lambda)=0 \implies \frac92-\frac{9\lambda}{2}=0 \implies \lambda=1.\]
\[P:\ -5x+8y-2z-13=0 \implies \vec a=(-5,8,-2),\ d=13.\]
\[|\vec a|^2=25+64+4=93.\]
\[\frac{13|\vec a|^2}{d}=\frac{13\times93}{13}=93.\] Answer the question to unlock