Let a hyperbola passes through the focus of the ellipse \( \frac{x^...

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Let a hyperbola passes through the focus of the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{16}=1 \). The transverse and conjugate axes of this
\( \mathrm{P} \) W 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,
(1) the equation of the hyperbola is \( \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \)
(2) the equation of the hyperbola is \( \frac{x^{2}}{9}-\frac{y^{2}}{25}=1 \)
(3) the focus of the hyperbola is \( (5,0) \)
(4) the vertex of the hyperbola is \( (5 \sqrt{3}, 0) \)
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