Let the length of the focal chord PQ of the parabola $$y^2 = 12x$$ be 15 units. If the distance of PQ from the origin is p, then $$10p^2$$ is equal to ______.

For $$y^2 = 4ax$$, $$a = 3$$.

Length of focal chord with inclination $$\theta$$ is $$4a \csc^2\theta$$.

$$12 \csc^2\theta = 15 \implies \sin^2\theta = \frac{12}{15} = \frac{4}{5}$$

Equation of focal chord passing through $$(a, 0) = (3, 0)$$:

$$y - 0 = \tan\theta(x - 3) \implies (\sin\theta)x - (\cos\theta)y - 3\sin\theta = 0$$

Distance $$p$$ from $$(0,0)$$:

$$p = \frac{|-3\sin\theta|}{\sqrt{\sin^2\theta + \cos^2\theta}} = 3\sin\theta$$

$$p^2 = 9\sin^2\theta = 9 \left(\frac{4}{5}\right) = \frac{36}{5}$$

 $$10p^2 = 10 \left(\frac{36}{5}\right) = \mathbf{72}$$.