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The difference between the focal dintences of any point on the hyperbola $\large\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=$$1$ is $24$ and the eccentricity is $2$. Then the equation of the hyperbola is

\[\begin{array}{1 1}(1)\frac{x^{2}}{144}-\frac{y^{2}}{432}=1&(2)\frac{x^{2}}{432}-\frac{y^{2}}{144}=1\\(3)\frac{x^{2}}{12}-\frac{y^{2}}{12\sqrt{3}}=1&(4)\frac{x^{2}}{12\sqrt{3}}-\frac{y^{2}}{12}=1\end{array}\]

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$SP-S'P =2a$
Eccentricity $e= 2$
$b^2= a^2(e^2-1)$
$\qquad= 12^2(4-1)$
$\qquad= 144 \times 3$
$b^2= 432$
Equation of the hyperbola is $\large\frac{x^{2}}{144}-\frac{y^{2}}{432}$$=1$
Hence 1 is the correct answer.
answered May 16, 2014 by meena.p

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