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(D) The difference between the focal distances of any point on a hyperbola is equal to the length of the transverse axis,which is $2a$. Given $2a = 6$

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$\pm \frac{4}{3}$

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(D) The difference between the focal distances of any point on a hyperbola is equal to the length of the transverse axis,which is $2a$. Given $2a = 6$,we find $a = 3$. The coordinates of the endpoints of the latus rectum are $(\pm ae, \pm \frac{b^2}{a})$. Since the point $(\sqrt{13}, k)$ is an endpoint of the latus rectum,we have $ae = \sqrt{13}$. Using the relation $c^2 = a^2 + b^2$,where $c = ae = \sqrt{13}$,we get $13 = a^2 + b^2$. Substituting $a = 3$,we have $13 = 3^2 + b^2$,which gives $13 = 9 + b^2$,so $b^2 = 4$. The $y$-coordinate of the endpoint of the latus rectum is $k = \pm \frac{b^2}{a}$. Substituting the values,$k = \pm \frac{4}{3}$.

The difference between the focal distances of any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $6$. If $(\sqrt{13}, k)$ is an endpoint of a latus rectum of this hyperbola,then $k=$

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