Step 1: Approach
Work with the quadratic in the ratio $y/x$.
Step 2: Quadratic
Divide the equation by $x^2$ and put $m=y/x$: $b m^2+2hm+a=0$. Its roots are the two slopes.
Step 3: Sum and product
Using Vieta: sum $=-\dfrac{2h}{b}$ and product $=\dfrac{a}{b}$.
Step 4: Compute
\[ \tan(\alpha+\beta)=\frac{-2h/b}{(b-a)/b}=\frac{2h}{a-b} \]
Step 5: Sanity test
Take $x^2-y^2=0$ (so $a=1,b=-1,h=0$): the lines are at 45 and 135 degrees. $\tan180^\circ=0$, and the formula gives $0$. Option (B) holds.
Final Answer:
$\tan(\alpha+\beta)=\dfrac{2h}{a-b}$, option (B).
\[ \boxed{\frac{2h}{a-b}} \]