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

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

The line 3x−49=1−3y9=z3 is perpendicular to the line of intersection of P1and P2

The acute angle between P1and P2 is 60°

If P3 is the plane passing through the point (4,2,−2) and perpendicular to the line of intersection of P1 and P2 , then the distance of the point (2,1,1) from the plane P3is 23

answer is C.

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Detailed Solution

The direction cosines of line of intersection of two planes is (a,b,c)a+2b+c=0,2a+b−c=0a1+2=b−1−2=c4−1⇒a3=b−3=c3⇒3,-3,3 are direction ratios of the line of intersection of planes.hence the option (1) is wrong3x−49=1−3y9=z3⇒⇒x−4/33=y−1/3−3=z3 , lines are parallelhence the option (2) is wrongAcute angle between the given two planes is =cos−12×1+1×2−1×166==cos−136=cos−112=60°Hence option (3) is correctEquation of the plane is given by x−4−y−2+z+2=0⇒x−y+z=0Distance from the point (2,1,1) to the above plane is =2−1+13=23Hence option (4) is corrent

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