Ratio of Areas of Triangles PQS and PQR

To find the ratio of the areas of the triangles $PQS$ and $PQR$, we first determine the coordinates of the points $P, Q, R,$ and $S$.

Step 1: Find the points of intersection $P$ and $Q$
The given equations are:

1. Circle: $x^2 + y^2 = 9$
2. Parabola: $y^2 = 8x$

Substituting $y^2 = 8x$ into the equation of the circle:
$x^2 + 8x - 9 = 0$
$(x + 9)(x - 1) = 0$

Since $x \ge 0$ for points on the parabola $y^2 = 8x$, we take:
$x = 1$

Substitute $x = 1$ into $y^2 = 8x$:
$y^2 = 8 \implies y = \pm 2\sqrt{2}$

Given that $P$ lies in the first quadrant and $Q$ lies in the fourth quadrant:
$P = (1, 2\sqrt{2}) \quad \text{and} \quad Q = (1, -2\sqrt{2})$

Step 2: Find the point of intersection $R$ of the tangents to the circle
The equation of the tangent to the circle $x^2 + y^2 = 9$ at point $P(1, 2\sqrt{2})$ is given by $T = 0$:
$x(1) + y(2\sqrt{2}) = 9 \implies x + 2\sqrt{2}y = 9$

Since $R$ is the intersection of the tangents with the $x$-axis, set $y = 0$:
$x_R = 9 \implies R = (9, 0)$

Step 3: Find the point of intersection $S$ of the tangents to the parabola
The equation of the tangent to the parabola $y^2 = 8x$ at point $P(1, 2\sqrt{2})$ is:
$y(2\sqrt{2}) = 4(x + 1)$

Since $S$ is the intersection of the tangents with the $x$-axis, set $y = 0$:
$4(x_S + 1) = 0 \implies x_S = -1 \implies S = (-1, 0)$

Step 4: Calculate the ratio of the areas of $\triangle PQS$ and $\triangle PQR$
Notice that both triangles share the common base $PQ$. The line passing through $P(1, 2\sqrt{2})$ and $Q(1, -2\sqrt{2})$ is the vertical line:
$x = 1$

The length of the common base is:
$PQ = 2\sqrt{2} - (-2\sqrt{2}) = 4\sqrt{2}$

The altitude $h_S$ of $\triangle PQS$ with respect to the line $x = 1$ is the horizontal distance from $S(-1, 0)$ to $x = 1$:
$h_S = |1 - (-1)| = 2$

The altitude $h_R$ of $\triangle PQR$ with respect to the line $x = 1$ is the horizontal distance from $R(9, 0)$ to $x = 1$:
$h_R = |9 - 1| = 8$

Since both triangles share the same base $PQ$, the ratio of their areas is equal to the ratio of their heights:
$\frac{\text{Area}(\triangle PQS)}{\text{Area}(\triangle PQR)} = \frac{h_S}{h_R} = \frac{2}{8} = \frac{1}{4}$

Hence, the ratio of the areas of the triangles $PQS$ and $PQR$ is $1 : 4$.

The correct option is C.