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Question:

Let P1: 2x+y-z=3 and P2: x+2y+z=2 be two planes. Then, which of the following statement(s) is (are) TRUE?

The line 3x=1+y/9=z/3 is perpendicular to the line of intersection of P1 and P2

The line of intersection of P1 and P2 has direction ratios 1,2,-1

The acute angle between P1 and P2 is 60o

If P3 is the plane passing through the point (4,2,-6) and perpendicular to the line of intersection of P1 and P2, then the distance of the point (2,1,1) from the plane P2 is 2√3

Solution:

The correct options are C and D.

C. of line of intersection (a, b, c) ⇒ 2a+b-c=0 a+2b+c=0
a+2 = b = -c/4 ⇒ D.C. is (1,-1,1)
(B) 3x=1+y/9=z/3 ⇒ x/1/3 = y-1/27 = z/9 ⇒ lines are parallel
(C) Acute angle between P1 and P2 = cos⁻¹(2×1+1×2-1×1/√6√6) = cos⁻¹(3/6) = cos⁻¹(1/2) = 60o
(D) Plane is given by (x-4)-(y-2)+(z+2)=0 ⇒ x-y+z=0
Distance of (2,1,1) from plane = |2-1+1|/√3 = 2/√3 = 2√3/3 ≠ 2√3