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JEE MainMathematicsThree Dimensional Geometry

A line L passing through the origin is equally inclined to the positive coordinate axes. L intersects the plane P₁: x 2A + y 2B + z 2C = d₁ at point Q₁ , and intersects the plane P₂: x A + y B + z C = d₂ at point Q₂ . Here A, B, C are the angles of a triangle, d₁ = 3 d₂ > 0 , and the distance OQ₁ = 4 OQ₂ (where O is the origin and Q₁, Q₂ lie on the same side of O ). The value of 16( A + B + C) is equal to _________.

Correct answer

22

Step-by-step solution

The line L passing through the origin and equally inclined to the positive coordinate axes has direction cosines ( 1 3 , 1 3 , 1 3 ) . Any point on L at a distance r from the origin has coordinates ( r 3 , r 3 , r 3 ) . Let OQ₁ = r₁ and OQ₂ = r₂ . We are given r₁ = 4r₂ . Since Q₁ lies on P₁ : r₁ 3 ( 2A + 2B + 2C) = d₁ Since Q₂ lies on P₂ : r₂ 3 ( A + B + C) = d₂ Dividing the first equation by the second, we get: r₁ r₂ 2A + 2B + 2C A + B + C = d₁ d₂ Substitute r₁/r₂ = 4 and d₁/d₂ = 3 : 4 2A + 2B + 2C A + B + C = 3 U

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