Geometric transformations - help

[TomBombadil]

I am preparing for an exam and have bumped into the following problem.
I have divided it in the following steps:
1) Identify the coordinates of the foci (F = (-sqrt(a^2 - b^2), 0), F' = (sqrt(a^2 - b^2), 0) ),
2) Calculate the coordinates of the centroids ( G of MFF' = ( x/3, y/3) ),
3) Identify locus of centroids ( This is were I don`t know how to continue),
4) Prove locus of centroids is a homotethy of the given ellipse. ( I do not know how this should be done, either).
Any help whatsoever would be greatly appreciated, thank you.

 

[Dr.Peterson]

I am preparing for an exam and have bumped into the following problem.
I have divided it in the following steps:
1) Identify the coordinates of the foci (F = (-sqrt(a^2 - b^2), 0), F' = (sqrt(a^2 - b^2), 0) ),
2) Calculate the coordinates of the centroids ( G of MFF' = ( x/3, y/3) ),
3) Identify locus of centroids ( This is were I don`t know how to continue),
4) Prove locus of centroids is a homotethy of the given ellipse. ( I do not know how this should be done, either).
Any help whatsoever would be greatly appreciated, thank you.

I think your step 2 shows step 4: the locus of centroids is obtained by scaling the ellipse by a factor of 1/3!

For step 3, suppose (X,Y) is a point in this locus. Then X = x/3 and Y = y/3; solve for x and y and substitute into the given equation.