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Question:

The point P is the intersection of the straight line joining the points Q (2,3,5) and R (1,-1,4) with the plane 5x - y - z = 1. If S is the foot of the perpendicular drawn from the point T (2,1,4) to QR, then the length of the line segment PS is 1√2

√2

2√2

2

1√2

Solution:

Direction ratio's of QR is (1,4,1). Co-ordinates of P ≡(4/3,1/3,13/3) Direction rations of point is (2,2,-1). Angle between QR and PT = cos⁻¹[(1×2+4×2+1×(-1))/√1²+4²+1²√2²+2²+(-1)²] = cos⁻¹[9/3√2×3] = cos⁻¹(1/√2) = 45°
PT = √(2-4/3)²+(1-1/3)²+(4-13/3)² = 1 ⇒ PS = TS = 1/√2