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JEE MainMathematicsThree Dimensional Geometry

Let a line L be given by the intersection of the planes 2x - y = 0 and 3x - z = 0 . If the mirror image of the point A(3, 4, 1) in the line L is ( , , ) , then the value of ^2 + ^2 + ^2 is equal to _______

Correct answer

26

Step-by-step solution

First, we find the symmetric equation of the line L . Let x = t . From the given planes, we have y = 2t and z = 3t . Thus, any general point on the line L can be written as M(t, 2t, 3t) , and the direction ratios of L are (1, 2, 3) . Let M be the foot of the perpendicular from A(3, 4, 1) to the line L . The direction ratios of the vector AM are (t - 3, 2t - 4, 3t - 1) . Since AM is perpendicular to the line L , the dot product of their direction ratios is zero: 1(t - 3) + 2(2t - 4) + 3(3t - 1) = 0 t - 3 + 4t - 8 +

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