Step-by-step Derivation

1. Parametric Coordinates on the Parabola The equation of the parabola is $y^2 = 4ax$. Let the endpoints of the focal chord $PQ$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$, where $t_1$ and $t_2$ are parameters.

2. Condition for a Focal Chord A chord is a focal chord if it passes through the focus, $S(a,0)$. The condition for the chord connecting points with parameters $t_1$ and $t_2$ to be a focal chord is: $t_1 t_2 = -1$

3. Point of Intersection of Tangents The equation of the tangent to the parabola $y^2 = 4ax$ at a point $(at^2, 2at)$ is given by $ty = x + at^2$. The tangent at $P$ is: $t_1y = x + at_1^2$. The tangent at $Q$ is: $t_2y = x + at_2^2$. The point of intersection of these two tangents is a standard result, given by the coordinates $(a t_1 t_2, a(t_1 + t_2))$.

4. Applying the Given Condition Since $PQ$ is a focal chord, we use the condition $t_1t_2 = -1$. The point of intersection becomes $(-a, a(t_1 + t_2))$. We are given that this point of intersection lies on the line $y = 2x + a$. Substituting the coordinates of the intersection point into the line's equation: $y = 2x + a$ $a(t_1 + t_2) = 2(-a) + a$ $a(t_1 + t_2) = -2a + a$ $a(t_1 + t_2) = -a$ Since $a > 0$, we can divide by $a$: $t_1 + t_2 = -1$

5. Calculating the Length of the Focal Chord The length of a focal chord $PQ$ can be found by summing the focal distances of its endpoints, $SP + SQ$. The focal distance of a point $(x,y)$ on the parabola $y^2 = 4ax$ is $x+a$. For point $P(at_1^2, 2at_1)$, the focal distance $SP = at_1^2 + a$. For point $Q(at_2^2, 2at_2)$, the focal distance $SQ = at_2^2 + a$. The length of the chord $PQ$ is: $L = SP + SQ = (at_1^2 + a) + (at_2^2 + a)$ $L = a(t_1^2 + t_2^2 + 2)$ We can express $t_1^2 + t_2^2$ in terms of $t_1+t_2$ and $t_1t_2$: $t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2t_1t_2$ From our previous steps, we know $t_1 + t_2 = -1$ and $t_1t_2 = -1$. Substituting these values: $t_1^2 + t_2^2 = (-1)^2 - 2(-1) = 1 + 2 = 3$ Now, substitute this result back into the formula for the length $L$: $L = a(3 + 2) = 5a$

   Alternative Method using Distance Formula: The length of the chord $PQ$ can also be calculated using the distance formula: $PQ^2 = (at_1^2 - at_2^2)^2 + (2at_1 - 2at_2)^2$ $PQ^2 = a^2(t_1^2 - t_2^2)^2 + 4a^2(t_1 - t_2)^2$ $PQ^2 = a^2[(t_1-t_2)(t_1+t_2)]^2 + 4a^2(t_1 - t_2)^2$ $PQ^2 = a^2(t_1-t_2)^2[(t_1+t_2)^2 + 4]$ We need to find $(t_1-t_2)^2$: $(t_1-t_2)^2 = (t_1+t_2)^2 - 4t_1t_2 = (-1)^2 - 4(-1) = 1+4 = 5$ Substituting the values: $PQ^2 = a^2(5)[(-1)^2 + 4] = a^2(5)(1+4) = 25a^2$ PQ = oxed{5a}

6. Conclusion The length of the focal chord $PQ$ is $5a$. This corresponds to option B.