If $$PQ$$ be a double ordinate of the parabola, $$y^2 = -4x$$, where $$P$$ lies in the second quadrant. If $$R$$ divides $$PQ$$ in the ratio 2 : 1, then the locus of $$R$$ is

We are given the parabola $$ y^2 = -4x $$. Since $$ PQ $$ is a double ordinate, it is perpendicular to the axis of the parabola. Given that the parabola is symmetric about the x-axis, $$ PQ $$ is horizontal. Let $$ P $$ be a point in the second quadrant, so its x-coordinate is negative and y-coordinate is positive. Let the coordinates of $$ P $$ be $$ (x_1, y_1) $$. Since $$ P $$ lies on the parabola, it satisfies the equation:

$$ y_1^2 = -4x_1 $$

Because $$ PQ $$ is a double ordinate, $$ Q $$ will have the same x-coordinate as $$ P $$ but the negative y-coordinate due to symmetry. Thus, the coordinates of $$ Q $$ are $$ (x_1, -y_1) $$.

Now, $$ R $$ divides $$ PQ $$ in the ratio 2:1. This means that the ratio $$ PR:RQ = 2:1 $$. Using the section formula, the coordinates of $$ R $$ are calculated as follows:

Let $$ R $$ have coordinates $$ (x_R, y_R) $$. The section formula for dividing the line segment joining $$ P(x_1, y_1) $$ and $$ Q(x_1, -y_1) $$ in the ratio $$ m:n = 2:1 $$ is:

$$ x_R = \frac{m \cdot x_Q + n \cdot x_P}{m + n} $$

$$ y_R = \frac{m \cdot y_Q + n \cdot y_P}{m + n} $$

Substituting $$ m = 2 $$, $$ n = 1 $$, $$ x_P = x_1 $$, $$ y_P = y_1 $$, $$ x_Q = x_1 $$, and $$ y_Q = -y_1 $$:

$$ x_R = \frac{2 \cdot x_1 + 1 \cdot x_1}{2 + 1} = \frac{2x_1 + x_1}{3} = \frac{3x_1}{3} = x_1 $$

$$ y_R = \frac{2 \cdot (-y_1) + 1 \cdot y_1}{3} = \frac{-2y_1 + y_1}{3} = \frac{-y_1}{3} = -\frac{y_1}{3} $$

So, $$ R $$ has coordinates $$ \left( x_1, -\frac{y_1}{3} \right) $$.

To find the locus of $$ R $$, we eliminate the parameters $$ x_1 $$ and $$ y_1 $$. Let $$ R $$ be denoted by $$ (h, k) $$, so:

$$ h = x_1 $$

$$ k = -\frac{y_1}{3} \quad \Rightarrow \quad y_1 = -3k $$

Since $$ P(x_1, y_1) $$ lies on the parabola $$ y^2 = -4x $$, substitute $$ x_1 = h $$ and $$ y_1 = -3k $$:

$$ (-3k)^2 = -4h $$

$$ 9k^2 = -4h $$

Replacing $$ h $$ and $$ k $$ with $$ x $$ and $$ y $$ respectively, the locus is:

$$ 9y^2 = -4x $$

Comparing with the given options:

A. $$ 3y^2 = -2x $$

B. $$ 9y^2 = 4x $$

C. $$ 9y^2 = -4x $$

D. $$ 3y^2 = 2x $$

The locus matches option C. Hence, the correct answer is Option C.