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JEE Advanced2016MathematicsThree Dimensional GeometryActual

Consider a pyramid O P Q R S located in the first octant x ≥ 0 , y ≥ 0 , z ≥ 0 with O , as origin, and O P and, O R along the x ‐ a x i s and the y ‐ a x i s , respectively. The base O P Q R of the pyramid is a square with O P = 3 . The point S is directly above the mid-point T of diagonal, O Q , such that, T S = 3 . Then

Options

  1. AThe acute angle between OQ and OS is π 3
  2. BThe equation of the plane containing the triangle OQS is x - y = 0
  3. CThe length of the perpendicular from P to the plane containing the triangle OQS is 3 2
  4. DThe perpendicular distance from O to the straight line containing RS is 15 2

Correct answer

B. The equation of the plane containing the triangle OQS is x - y = 0

Step-by-step solution

O 0,0 , 0 → Origin P 3,0 , 0 → on x axis R 0,3 , 0 → on y axis Q 3,3 , 0 T 3 2 , 3 2 , 0 , 5 3 2 , 3 2 , 3 Given OP = OR = 3 and OPQR is a square ⇒ O Q = 3 2 ⇒ O T = 3 2 and S T = 3 Let θ be a angle between OQ & OS Using Δ S O T , tan ⁡ θ = S T O T = 2 ⇒ θ = tan - 1 ⁡ 2 Clearly, equation of plane containing triangle OQS is x - y = 0 as O 0,0 , 0 , Q 3,3 , 0 , S 3 2 , 3 2 , 3 lies on it let ax + by + cz = d 0,0 , 0 ⇒ d = 0 3,3 , 0 ⇒ a + b = 0 3 2 , 3 2 , 3 ⇒ 3 a 2 + 3 b 2 + 3 c = 0 ⇒ c = 0 ⇒ b = - a x - y = 0 Also,

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