- #1
The idea with the string is that you pull it so that it is taut.So you mean that suppose I take the same thumb tacks but I pull the string with a bit lesser force. Now I pull the string till as much as it can stretch. Now in the first case if the sum of distances of all the points is 12. In the latter case when i pull the string hard also the sum of all the distances will be 12. Am I right? But for a point that is not the focus the sum of distances will vary when the shape changes. Am I right?
Your sentence above is very confused, as it implies that "the foci" is the locus of some points. That's not true.
The idea with the string is that you pull it so that it is taut.
Here is a picture of what jedishrfu is talking about. The foci (plural of focus) are at F1and F2 and P is an arbitrary point on the ellipse. For an ellipse, the sum of the lengths of the segments F1P and PF2 is a constant. As we move around the ellipse to the right, the segment F1P gets longer and the segement PF2 gets shorter, but the sum F1P + PF2 remains the same.
If you move one or both of the thumbtacks, which changes the location of the foci, you get a different ellipse.