The hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ passes through the point $(2, 3)$ and has the eccentricity $2$ . Then the transverse axis of the hyperbola has the length

1

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Solution

The correct option is C $2$
Given hyperbola is, $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
It passes through $(2, 3)$
$\frac{4}{a^{2}} - \frac{9}{b^{2}} = 1 . . (1)$
Also eccentricity is $2$ ,
$\Rightarrow e^{2} = 1 + \frac{b^{2}}{a^{2}} = 4 \Rightarrow \frac{b^{2}}{a^{2}} = 3 . . (2)$
Solving (1) and (2) we get $a = 1, b = \sqrt{3}$
Hence, length of transverse axis is $= 2 a = 2$

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The hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ passes through the point $(2, 3)$ and has the eccentricity $2$ . Then the transverse axis of the hyperbola has the length