How Does a Point Charge Affect Induced Charges on Grounded Infinite Plates?

[pcm]

Homework Statement

two infinite conducting plates 1 and 2(both grounded or connected by a wire so that their potential is same) are separated by a distance l.
A point charge q is located between the plates at a distance x from plate 1.
find the charges induced on each plate.

Homework Equations

The Attempt at a Solution

the book says that if the charge q is imagined to be uniformly spread over the plane passing through that charge and parallel to the plates , the charges induced on plates remains unchanged, only their distribution changes.
i can solve the problem if i assume it.i will assume a certain charge distribution and field can be easily found out and using that i can find change in potential change in going from either plates to the middle plate.and that should be equal.
but can anyone tell me from where that assumption came from.

 

[pcm]

if the charge q is imagined to be uniformly spread over the plane passing through that charge and parallel to the plates , the charges induced on plates remains unchanged, only their distribution changes.

can anyone explain that...

 

[pcm]

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