jmc

In general, for the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ the asymptotes are $\frac{x}{a}=\pm \frac{y}{b},$ or $y=\pm \frac{b}{a}x.$ Therefore, the asymptotes of the first hyperbola are $y=\pm \frac{3}{2}x.$

For the hyperbola $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1,$ the asymptotes are $\frac{y}{a}=\pm \frac{x}{b},$ or $y=\pm \frac{a}{b}x.$ Therefore, the asymptotes of the second hyperbola are $y=\pm \frac{3\sqrt{2}}{\sqrt{N}}x.$

For the two hyperbolas to have the same asymptotes, we must have $\frac{3}{2}=\frac{3\sqrt{2}}{\sqrt{N}}.$ Solving for $N$ gives $N=\boxed{8}.$