Line segment $W X$ is the radius of circle $X$, and line segment $Z Y$ is the radius of circle $Y$. Points $W, X, C, Y$, and $Z$ are all on line segment $W Z$.

What is the area of circle $C$, which passes though points $W$ and $Z$ ?
$81 \pi$ units $^{2}$
$164 \pi$ units $^{2}$
$324 \pi$ units $^{2}$
$1296 \pi$ units $^{2}$

Step 1 ：Let's denote the radius of circle X as \(r_x\) and the radius of circle Y as \(r_y\). The radius of circle C is the sum of the radii of circle X and circle Y, so \(r_c = r_x + r_y\).

Step 2 ：The area of circle C is \(\pi (r_x + r_y)^2\).

Step 3 ：We can observe that the area of circle C is the sum of the areas of circles X and Y plus the area of the rectangle formed by \(r_x\) and \(r_y\).

Step 4 ：The area of the rectangle is \(2 \pi r_x r_y\), and the areas of circles X and Y are \(\pi r_x^2\) and \(\pi r_y^2\) respectively.

Step 5 ：Therefore, the area of circle C is \(\pi r_x^2 + \pi r_y^2 + 2 \pi r_x r_y = \pi (r_x + r_y)^2\).

Step 6 ：Since the area of circle C is the sum of the areas of circles X and Y plus the area of the rectangle formed by \(r_x\) and \(r_y\), the area of circle C must be greater than the areas of circles X and Y.

Step 7 ：Therefore, the area of circle C must be \(1296 \pi\) units \(^{2}\), which is the largest option given in the question.

Step 8 ：Final Answer: \(\boxed{1296 \pi}\) units \(^{2}\)