A series of concentric ellipses E 1 , E 2 ,......,E n are drawn such that E n touches the extremities of the major axis of E n –1 and the foci of E n coincide with the extremities of minor axis of E n – 1 . If the eccentricity of the ellipses is independent of n, then the value of the eccentricity is
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Sol. The figure shows two ellipses E n – 1 and E n .
The eccentricity is given to be independent of n, implies that the ratio of minor axis to the major axis, is same for all the ellipses.
For ellipse E n – 1 , let
minor axis = b, major axis = a
For ellipse E n , we have
minor axis = a, major axis = 
= 
[ B is the focus of E n ]
assuming e to be the eccentricity. Thus, we have
= 
⇒ e = 
= 1 – e 2 ⇒ e 2 + e – 1 = 0
gives e = 
[ e must be + ve]
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