Length of Latus Rectum

Given that $$y = 2x$$, substitute this relationship into the equation of the parabola:

$$(2x)^2 = 8x \implies 4x^2 = 8x$$

Since $$P$$ is a non-trivial point away from the origin ($$x \neq 0$$), dividing by $$4x$$ gives:

$$x = 2 \implies y = 2(2) = 4$$

Therefore, the point is $$P(2, 4)$$.

Comparing $$y^2 = 8x$$ with the standard equation $$y^2 = 4ax$$, we get:

$$4a = 8 \implies a = 2$$

The focus of this parabola is located at $$S(a, 0) = (2, 0)$$.

Notice that the x-coordinate of point $$P$$ ($$x = 2$$) is perfectly identical to the x-coordinate of the focus $$S$$ ($$x = 2$$).

This indicates that the line passing through $$P$$ and $$S$$ is completely vertical, meaning it is perpendicular to the axis of symmetry (the x-axis).

A focal chord that stands perpendicular to the axis of a parabola is defined as its latus rectum.

The length of the latus rectum for any standard parabola is given directly by the coefficient $$4a$$:

$$\text{Length} = 4a = 8$$