The foci of the ellipse and the hyperbola coincide. Then the value of is
Explanation for correct option:
Step-1 : Formula for eccentricity for both curves:
Given: The foci of the ellipse and the hyperbola coincide.
We know that ,
Formula for the eccentricity of hyperbola :
Formula for the eccentricity of ellipse :
Step-2 : Find the foci of the given conic section :
Hyperbola equation becomes,
Now ,Comparing the equation (1) with the standard equation of hyperbola .
So , we have
Now, Eccentricity of the hyperbola :-
Assume is the eccentricity of hyperbola.
We know that , coordinates of foci of standard hyperbola are
Here ,
So , foci is
Foci is .
Step-3 : Equating foci of the both curves:
Now , Assume is the eccentricity of ellipse.
Therefore the foci of the ellipse and hyperbola are at same points.
So,
Hence, the value of is ,
Therefore, option (C) is the correct answer.