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**B1 Algebra**

Here the 10 cm string has been pulled as far as possible to the right.

Point P is the far-right point of the ellipse, point C is the centre of the ellipse.

We can argue as follows:

u + u + v + v = 10, so CP = u + v = 5,

so the ellipse is 2x5 = 10 units wide, regardless of the distance between A and B.

This algebraic 'proof' is elegant, but is it illuminating? We look at some geometric arguments next.