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10-2. Parabola, Ellipse, Hyperbola
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The equation of the hyperbola whose conjugate axis is $5$ and the distance between the foci is $13$, is
A
$25{x^2} - 144{y^2} = 900$
B
$144{x^2} - 25{y^2} = 900$
C
$144{x^2} + 25{y^2} = 900$
D
$25{x^2} + 144{y^2} = 900$
Solution
(a) Conjugate axis is $5$ and distance between foci = $13$
==> $2b = 5$ and $2ae = 13$.
Now, also we know for hyperbola
${b^2} = {a^2}({e^2} – 1)$
==> $\frac{{25}}{4} = \frac{{{{(13)}^2}}}{{4{e^2}}}({e^2} – 1)$
==> $\frac{{25}}{4} = \frac{{169}}{4} – \frac{{169}}{{4{e^2}}}$ or ${e^2} = \frac{{169}}{{144}}$
==> $e = \frac{{13}}{{12}}$
or $a = 6,\,b = \frac{5}{2}$ or hyperbola is $\frac{{{x^2}}}{{36}} – \frac{{{y^2}}}{{25/4}} = 1$
==> $25{x^2} – 144{y^2} = 900$.
Standard 11
Mathematics
