# Sum of the distances of any number of fixed point from a variable plane is zero. Show that the plane passes through a fixed point

2086 Points
4 years ago
we know that the distance is always positive.
let the given fixed points be P1, P2, …......., Pn at distances d1, d2,........, dn from the plane.
we know that di is greater than equal to 0.
so ∑di is greater than equal to 0.
but ∑di is given to be 0.
hence the equality holds iff di= 0 for all i belonging to {1, 2, …....., n}.
this means that all these points lie on the plane (since their distance from plane has been shown to be 0).
now, let us assume that these points are non collinear. that would imply that a unique plane passes through them. However, it is given that the plane can be variable. Hence the points are bound to be collinear. Let them lie on line L.
hence, any variable plane passing thru all these points would be a plane containing L, and many different such planes can exist due to possibility of rotation of planes about L.
so, the plane passes thru not just through one but all the fixed points P1, P2,...., Pn.
kindly approve :))
Vikas TU
14149 Points
4 years ago
Dear student
Good Luck
Rishi Sharma
4 years ago
Dear Student,

we know that the distance is always positive.
let the given fixed points be P1, P2, …......., Pn at distances d1, d2,........, dn from the plane.
we know that di is greater than equal to 0.
so∑diis greater than equal to 0.
but∑diis given to be 0.
hence the equality holds iff di= 0 for all i belonging to {1, 2, …....., n}.
this means that all these points lie on the plane (since their distance from plane has been shown to be 0).
now, let us assume that these points are non collinear. that would imply that a unique plane passes through them. However, it is given that the plane can be variable. Hence the points are bound to be collinear. Let them lie on line L.
hence, any variable plane passing thru all these points would be a plane containing L, and many different such planes can exist due to possibility of rotation of planes about L.
so, the plane passes thru not just through one but all the fixed points P1, P2,...., Pn.

Thanks and Regards