What is the distance between the two foci of the hyperbola 25x^2 -...

The given equation of the hyperbola is $25x^2 - 75y^2 = 225$.

Dividing the entire equation by $225$, we get:

$\dfrac{x^2}{9} - \dfrac{y^2}{3} = 1$

Comparing this with the standard equation of a hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, we have $a^2 = 9$ and $b^2 = 3$.

The relation between $a$, $b$, and the distance from the center to a focus $c$ is given by $c^2 = a^2 + b^2$.

Substituting the values, we get:

$c^2 = 9 + 3 = 12$

$c = \sqrt{12} = 2\sqrt{3}$

The distance between the two foci is $2c$.

Distance $= 2(2\sqrt{3}) = 4\sqrt{3}$ units.

Answer: $4\sqrt{3}$ units