**Solution:**

To find the equation of the hyperbola, we need to determine the coordinates of its center and the equations of its asymptotes.

**Finding the Center:**

Given that the center of the hyperbola lies on the line y = 3x, we can substitute y = 3x into the general equation of the hyperbola to find the coordinates of the center.

The general equation of a rectangular hyperbola is given by:

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Substituting y = 3x into the equation, we get:

$\frac{x^2}{a^2} - \frac{(3x)^2}{b^2} = 1$

Simplifying the equation, we have:

$\frac{x^2}{a^2} - \frac{9x^2}{b^2} = 1$

To eliminate the fraction, we can multiply the entire equation by a^2b^2:

$b^2x^2 - 9a^2x^2 = a^2b^2$

Factoring out x^2, we get:

$(b^2 - 9a^2)x^2 = a^2b^2$

Comparing the equation with the standard form of a quadratic equation, we can determine that the coefficient of x^2 is zero:

$b^2 - 9a^2 = 0$

Solving for b^2, we find:

$b^2 = 9a^2$

Therefore, the coordinates of the center (h, k) are given by:

(h, k) = (0, 0)

**Finding the Asymptotes:**

The equation of the asymptotes of a rectangular hyperbola is given by:

$\frac{x}{a} \pm \frac{y}{b} = 0$

Substituting the values of a and b, we have:

$\frac{x}{a} \pm \frac{y}{3a} = 0$

To find the other asymptote, we need to consider the equation given in the question: x + y + k = 0

Comparing this equation with the equation of the asymptotes, we can determine that the slope of the other asymptote is -1.

Therefore, the equation of the other asymptote is given by:

y = -x

In conclusion, the equation of the hyperbola is $\frac{x^2}{a^2} - \frac{y^2}{9a^2} = 1$, and the equations of the asymptotes are y = 3x and y = -x.