A straight line drawn from the point P ( 1 , 3 , 2 ) , parallel to the line 1 x − 2 = 2 y − 4 = 1 z − 6 , intersects the plane L 1 : x − y + 3 z = 6 at the point Q . Another straight line whic

Question

A straight line drawn from the point P ( 1 , 3 , 2 ) , parallel to the line 1 x − 2 ​ = 2 y − 4 ​ = 1 z − 6 ​ , intersects the plane L 1 ​ : x − y + 3 z = 6 at the point Q . Another straight line which passes through Q and is perpendicular to the plane L 1 ​ intersects the plane L 2 ​ : 2 x − y + z = − 4 at the point R . Then which of the following statements is (are) TRUE?

TestHub · Accepted Answer

​ line : 1 x − 1 ​ = 2 y − 3 ​ = 1 z − 2 ​ ( x , y , z ) = ( λ + 1 , 2 λ + 3 , λ + 2 ) ​ Put in L 1 ​ : x − y + 3 z = 6 ​ ( λ + 1 ) − ( 2 λ + 3 ) + 3 ( λ + 2 ) = 6 2 λ = 2 ⇒ λ = 1 Q = ( 2 , 5 , 3 ) ​ line : 1 x − 2 ​ = − 1 y − 5 ​ = 3 z − 3 ​ ( x , y , z ) = ( t + 2 , 5 − t , 3 t + 3 ) Put in L 2 ​ : 2 x − y + z = − 4 ​ 2 ( t + 4 ) − ( 5 − t ) + ( 3 t + 3 ) = − 4 6 t = − 6 ⇒ t = − 1 R = ( 1 , 6 , 0 ) ​ ​ Perimeter = 6 ​ + 13 ​ + 11 ​ Centroid = ( 3 4 ​ , 3 14 ​ , 3 5 ​ ) ​

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