Length of Focal Chord for Parabola Given Tangent Intersection Line

To find the length of the focal chord $PQ$ of the parabola $y^2 = 4ax$, let the parametric coordinates of the endpoints $P$ and $Q$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$, respectively.

Since $PQ$ is a focal chord, the parameters $t_1$ and $t_2$ satisfy the standard property: $t_1 t_2 = -1$

The point of intersection $T(x_0, y_0)$ of the tangents drawn to the parabola at points $P$ and $Q$ is given by: $x_0 = a t_1 t_2$ $y_0 = a (t_1 + t_2)$

Substituting $t_1 t_2 = -1$ into the expression for $x_0$: $x_0 = -a$

It is given that the point $T(-a, a(t_1 + t_2))$ lies on the line $y = 2x + a$. Substituting the coordinates of $T$ into this line equation gives: $a(t_1 + t_2) = 2(-a) + a$ $a(t_1 + t_2) = -a$

Since $a > 0$, we can divide both sides by $a$: $t_1 + t_2 = -1$

Now, using the algebraic identity $(t_1 - t_2)^2 = (t_1 + t_2)^2 - 4t_1t_2$, we substitute the values $t_1 + t_2 = -1$ and $t_1 t_2 = -1$: $(t_1 - t_2)^2 = (-1)^2 - 4(-1) = 1 + 4 = 5$

The length of a focal chord of a parabola $y^2 = 4ax$ with endpoints defined by parameters $t_1$ and $t_2$ is given by: $\text{Length}(PQ) = a(t_1 - t_2)^2$

Substituting $(t_1 - t_2)^2 = 5$ into the length formula: $\text{Length}(PQ) = 5a$

Thus, the correct option is B.