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JEE Main202227 Jun 2022Morning ShiftMathematicsThree Dimensional GeometryActual

Let the mirror image of the point a , b , c with respect to the plane 3 x - 4 y + 12 z + 19 = 0 be a - 6 , β , γ . If a + b + c = 5 , then 7 β - 9 γ is equal to ______.

Correct answer

0

Step-by-step solution

Given the mirror image of P a , b , c is P ' a - 6 , β , γ The mid-point M = a - 3 , β + b 2 , γ + c 2 Since M lies on 3 x - 4 y + 12 z + 19 = 0 ⇒ 3 a - 9 - 4 β + b 2 + 12 γ + c 2 + 19 = 0 ⇒ 6 a - 18 - 4 β - 4 b + 12 γ + 12 c + 38 = 0 ⇒ 6 a - 4 b + 12 c - 4 β + 12 γ + 20 = 0             ⋯ 1 Since P P ' is parallel to normal of the plane then 6 3 = b - β - 4 = c - γ 12 ⇒ b = β - 8 , &#

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