Expert: Azeem Hussain - 11/25/2007

Question
Hi There;
I am Mohammed Babaee, an Expert in Windows Mobile Devices at AllExperts.
I have one question:
How can we prove that if the Eccentricity of an ellipse equals to 1 (e=1), we will have a line.

Thanks in advance

Answer
Hi Mohammed!

I would use logic to show this.  If e=0, then the ellipse is really a circle, for both foci are superimposed.  As the eccentricity increases from 0 to 1, the circle flattens out, forming an ellipse.

The formula for calculating eccentricity is e=(1-(b^2/a^2))^0.5.  When semi-major axis (a) and semi-minor axis (b) are equivalent, eccentricity equals 0.  In order to have an eccentricity of 1, the quotient of the squares of the semi-major and semi-minor must be 0.  The only way to divide and have 0 is to divide by infinity.  Therefore, when the semi-major axis is infinity, regardless of what the semi-minor axis is, eccentricity equals 1.  At this point, it is impossible to distinguish one side of the ellipse from the other; we have a line.

Please note that I am not an expert on ellipses, but this is what I know and it is the way I would prove it, as you have asked me.

I hope this was helpful and I wish you all the best in your future answers,
Azeem