Let 
and 
be the eccentricities of the ellipse 
and the hyperbola 
, respectively. If 
and 
, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:
Text Solution
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Let's find the eccentricities of the given ellipse and hyperbola, and then determine the eccentricity of an ellipse that passes through all four foci.
Find 
for the Ellipse
The equation of the ellipse is:
The eccentricity 
is given by:
Find 
for the Hyperbola
The equation of the hyperbola is:
The eccentricity 
is given by:
Using the Product 
Given:
Thus:
Expanding gives:
Simplifying:
Thus:
Determine Eccentricities 
and 
Substitute 
:
For the ellipse:
For the hyperbola:
Find the Eccentricity of the New Ellipse
The new ellipse's equation is:
The eccentricity 
is:
Thus, the eccentricity of the ellipse that passes through all four foci is 
.
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