Step 1: Direct slopes:
Divide the equation by $x^2$: $b m^2 + 2h m + a = 0$ where $m = y/x$.
Step 2: Discriminant form:
$m = \dfrac{-h \pm \sqrt{h^2 - ab}}{b}$. With $h^2 = \tfrac43 ab$, $h^2 - ab = \tfrac13 ab = \tfrac{h^2}{4}$, so $\sqrt{h^2 - ab} = h/2$.
Step 3: Slopes:
$m = \dfrac{-h \pm h/2}{b}$, giving $-\dfrac{h}{2b}$ and $-\dfrac{3h}{2b}$. The ratio is $1:3$, option (D).
Final Answer:
The slopes are in the ratio 1 : 3.
\[ \boxed{\text{(D) }1:3} \]