Can an ellipse with fixed semi-axis have different values of eccentricity?

Question

Warning: this is probably a ridiculous question but here goes...

Can an ellipse with a semi-major axis $a$ take on different values of eccentricity $e$? I have seen various places where it seems to assume that the above is implicitly true, but I cannot visualise it. How can I convince myself that it is true, if it is true?

Answer 1

If the semi-major axis is $a$ and the semi-minor axis is $b$, then the eccentricity is given by $$ e=\sqrt{1-\frac{b^2}{a^2}} $$ Thus, by leaving the semi-major axis alone and varying the semi-minor axis, we can get any eccentricity between $0$ and $1$.

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Answer 2

The eccentricity depends on the length on the length of the minor axis, too: when $b=a$ the ellipse is a circle, so its eccentricity is zero, and when $b$ approaches zero the eccentricity approaches $1$, all without changing the length of the major axis.