Suppose $$AB$$ is a focal chord of the parabola $$y^2 = 12x$$ of length $$l$$ and slope $$m < \sqrt{3}$$. If the distance of the chord $$AB$$ from the origin is $$d$$, then $$ld^2$$ is equal to ______

$$y^2 = 12x \implies a=3$$. Focus is $$(3,0)$$.

Chord length $$l = 4a \csc^2 \theta = 12(1 + \frac{1}{m^2})$$.

Distance $$d$$ from $$(0,0)$$ to line $$y = m(x-3)$$: $$d = \frac{|-3m|}{\sqrt{m^2+1}}$$.

$$ld^2 = \left[ \frac{12(m^2+1)}{m^2} \right] \cdot \frac{9m^2}{m^2+1} = 108$$.

Answer: 108