Locus of Point Dividing Line Segment from Origin to Parabola

To find the locus of the point $P$, let the coordinates of $P$ be $(h, k)$.

Let the origin be $O(0, 0)$ and the point on the parabola be $Q(x, y)$.

Since the point $P(h, k)$ divides the directed line segment from $O(0, 0)$ to $Q(x, y)$ in the ratio $1 : 3$, using the section formula, we have:
$h = \frac{1 \cdot x + 3 \cdot 0}{1 + 3} = \frac{x}{4} \implies x = 4h$
$k = \frac{1 \cdot y + 3 \cdot 0}{1 + 3} = \frac{y}{4} \implies y = 4k$

The point $Q(x, y)$ lies on the parabola given by:
$y^2 = 4x$

Substituting $x = 4h$ and $y = 4k$ into the equation of the parabola:
$(4k)^2 = 4(4h)$
$16k^2 = 16h$
$k^2 = h$

Replacing $(h, k)$ with $(x, y)$ gives the locus of $P$:
$y^2 = x$

Therefore, the correct option is C.