A plane meets the coordinate axes at points A, B and C, such that the centroid of a triangle ABC is (2, - 2 3 , 1 2 ) . The perpendicular distance from the origin to this plane is...

Step-by-step solution

Let the coordinates of points A, B, and C on the coordinate axes be (a, 0, 0) , (0, b, 0) , and (0, 0, c) respectively. The equation of the plane passing through these points is x a + y b + z c = 1 . The centroid of the triangle ABC is given by ( a 3 , b 3 , c 3 ) . Comparing this with the given centroid (2, - 2 3 , 1 2 ) , we obtain: a 3 = 2 a = 6 b 3 = - 2 3 b = -2 c 3 = 1 2 c = 3 2 Substituting the values of a , b , and c in the equation of the plane: x 6 + y -2 + z 3 2 = 1 x 6 - y 2 + 2z 3 = 1 Multiplying the e