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Question

# If three planes P1≡2x+y+z−1=0, P2≡x−y+z−2=0 and P3≡αx−y+3z−5=0 intersect each other at point P on XOY plane and at point Q on YOZ plane, where O is the origin, then identify the correct statement(s)?

A
The length of projection of PQ on xaxis is 1. unit
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B
Straight line perpendicular to plane P3 and passing through P is x14=y+11=z3.
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C
Centroid of the triangle OPQ is (13,12,12)
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D
The value of α is 4.
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Solution

## The correct option is D The value of α is 4.Three planes meet at two points it means they have infinitely many solutions. ∴∣∣ ∣∣2111−11α−13∣∣ ∣∣=0 ⇒2(−3+1)−1(3−α)+1(−1+α)=0 ⇒α=4 Thus, equation of planes are P1:2x+y+z=1 P2:x−y+z=2 P3:4x−y+3z=5 Since point P lies on XOY plane ∴ Put z=0 in the above planes ⇒x=1,y=−1 ∴P(1,−1,0) and point Q lies on YOZ plane ∴ Put x=0 in the above planes ⇒z=32,y=−12 ∴Q(0,−12,32) Straight line perpendicular to plane P3 and passing through P is x−14=y+1−1=z3. We have, −−→PQ=^i+12^j+32^k Now, the length of projection of −−→PQ on x−axis is ∣∣ ∣∣(^i+12^j+32^k)⋅(±^i|±^i|)∣∣ ∣∣ 1 unit. Centroid of △OPQ is (13,−12,12).

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