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JEE MainMathematicsThree Dimensional Geometry

Let P be the orthogonal projection of the origin on the plane x+y+z=d . It is given that P also lies on the planes x ^2 A + y ^2 B + z ^2 C = 21 and x ^2 A + y ^2 B + z ^2 C = 15 , where A, B, C are the angles of a triangle. The value of 24(1 + A B C) is equal to _________.

Correct answer

21

Step-by-step solution

The orthogonal projection of the origin on the plane x+y+z=d must lie on the normal to the plane passing through the origin. The equation of this normal line is x=y=z . Let the coordinates of P be (k, k, k) . Since P lies on the given planes, we substitute x=y=z=k into their equations: k( ^2 A + ^2 B + ^2 C) = 21 k( ^2 A + ^2 B + ^2 C) = 15 Adding the two equations, we get: k( ^2 A + ^2 A + ^2 B + ^2 B + ^2 C + ^2 C) = 21 + 15 k(1 + 1 + 1) = 36 3k = 36 k = 12 Substituting k = 12 back into the second equation: 12( ^

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