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JEE Advanced2025MathematicsThree Dimensional GeometryActual

Let L₁ be the line of intersection of the planes given by the equation- 2 x+3 y+z=4 and x+2 y+z=5 . Let L₂ be the line passing through the point P(2,-1,3) and parallel to L₁ . Let M denote the plane given by the equation- 2 x+y-2 z=6 Suppose that the line L₂ meets the plane M at the point Q . Let R be the foot of the perpendicular drawn from P to the plane M . Then which of the following statements is (are) TRUE?

Options

  1. AThe length of the line segment P Q is 9 3
  2. BThe length of the line segment Q R is 15
  3. CThe area of P Q R is 3 2 234
  4. DThe acute angle between the line segments P Q and P R is ⁻¹ ( 1 2 3 )

Correct answer

A. The length of the line segment P Q is 9 3

Step-by-step solution

Let L₁: r = a +t b b = | array ccc i & j & k 2 & 3 & 1 1 & 2 & 1 array |= i (1)- j (1)+ k (1) Dr's of L₁: aligned & L₁: x+1 1 = y-0 -1 = z-6 1 & & L₂: x-2 1 = y+1 -1 = z-3 1 = & M: 2 x+y-2 z-6=0 aligned Let point on L ₂( +2,- -1, +3) aligned & 2( +2)- -1-2 -6-6=0 & 2 +4-3 -13=0 & =-9 & Q(-7,8,-6) aligned Line PR: x-2 2 = y+1 1 = z-3 -2 = R (2 +2, -1,-2 +3) Put in plane aligned & 2(2 +2)+ -1-2(-2 +3)-6=0 & 4 +4+ -1+4 -6-6=0 & 9 -9=0 =1 & R (4,0,1) & P (2,-1,3) & Q (-7,8,-6) & PQ = 81+81+81 =9 3 & Q (-7,8,-6) & R (4,

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