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(A) The standard equation of a hyperbola with centre at the origin and transverse axis along the $x$-axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.G

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$\frac{2}{\sqrt{3}}$

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(A) The standard equation of a hyperbola with centre at the origin and transverse axis along the $x$-axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.Given that the length of the transverse axis is $2a = 4$,we have $a = 2$,so $a^2 = 4$.The equation becomes $\frac{x^2}{4} - \frac{y^2}{b^2} = 1$.Since the hyperbola passes through the point $(4, 2)$,we substitute these values into the equation:$\frac{4^2}{4} - \frac{2^2}{b^2} = 1$$\frac{16}{4} - \frac{4}{b^2} = 1$$4 - \frac{4}{b^2} = 1$$3 = \frac{4}{b^2}$$b^2 = \frac{4}{3}$.The eccentricity $e$ is given by $e = \sqrt{1 + \frac{b^2}{a^2}}$.Substituting the values of $a^2$ and $b^2$:$e = \sqrt{1 + \frac{4/3}{4}} = \sqrt{1 + \frac{1}{3}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}$.

$A$ hyperbola has its centre at the origin,passes through the point $(4, 2)$ and has a transverse axis of length $4$ along the $x$-axis. Then the eccentricity of the hyperbola is

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