A plane meets the coordinate axes at A, B and C respectively such that the centroid of ∆ABC is (1, -2, 3). Find the equation of the plane.
Given :
X-intercept = A
Y-intercept = B
Z-intercept = C
Centroid of ∆ABC = (1, -2, 3)
To Find : Equation of a plane
Formulae :
1) Centroid Formula :
For ∆ABC if co-ordinates of A, B & C are
A = (x1, x2, x3)
B = (y1, y2, y3)
C = (z1, z2, z3)
Then co-ordinates of the centroid of ∆ABC are
2) Equation of plane :
Equation of the plane making a, b & c intercepts with X, Y & Z axes respectively is
As the plane makes intercepts at points A, B & C on X, Y & Z axes respectively, let co-ordinates of A, B, C be
A = (a, 0, 0)
B = (0, b, 0)
C = (0, 0, c)
By centroid formula,
The centroid of ∆ABC is given by
But, Centroid of ∆ABC = (1, -2, 3) …… given
Therefore, a = 3, b = - 6, c = 9
Therefore,
X-intercept = a = 3
Y-intercept = b = - 6
Z-intercept = c = 9
Therefore, equation of required plane is