Let a hyperbola passes through the focus of the ellipse . The transverse and conjugate axes of this hyperbola coincide with the major and minor axes of the given ellipse. Also, the product of the eccentricities of the given ellipse and hyperbola is 1. Then,
For the given ellipse
we have
Hence, the eccentricity of the hyperbola is 5/3.
Let the hyperbola be
Then
Therefore, the equation of the hyperbola is
As it passes through (3, 0), we get A2=9 and B2 =16' The equation is
The foci of the hyperbola are (ae,0) = (5,0).
The vertex of the hyperbola is (3, 0)