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JEE MainMathematicsThree Dimensional Geometry

Let a plane P pass through the origin O(0,0,0) and the points A(3, -1, 2) and B(4, 5, -3) . If a point C( , 3, ) lies on the plane P , where and are integers such that | | + | | is minimum, then the value of ^2 + ^2 is equal to

Correct answer

101

Step-by-step solution

The normal vector to the plane P is given by the cross product of vectors OA and OB . n = OA OB = vmatrix i & j & k 3 & -1 & 2 4 & 5 & -3 vmatrix n = i (3 - 10) - j (-9 - 8) + k (15 + 4) = -7 i + 17 j + 19 k The equation of the plane passing through the origin is -7x + 17y + 19z = 0 . Since the point C( , 3, ) lies on the plane, we substitute its coordinates into the equation: -7 + 17(3) + 19 = 0 7 - 19 = 51 We need to find integer values of and that satisfy this linear Diophantine equation and minimize | | + | | .

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