To solve this problem, we need to interpret the graphical information of the circle $C$ and find the equation of a tangent line.

Step-by-Step Analysis:

1. Circle's Center and Radius:

   • The center of the circle $O$ is clearly shown at $(0, 0)$, which is the origin.
   • From the diagram, the circle passes through the point $(-8, 0)$, meaning the radius $r$ of the circle is 8.

2. Equation of the Circle:

   • The general equation of a circle with center $(h, k)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$ Since the center is at $(0, 0)$ and the radius is 8, the equation of the circle is: $x^2 + y^2 = 64$

3. Tangent Line to the Circle:

   • The question asks for the equation of a line tangent to the circle.
   • A vertical line that is tangent to the circle will be of the form $x = a$.
   • Since the radius of the circle is 8, the vertical tangent lines occur at $x = -8$ (left side) and $x = 8$ (right side).

4. Options for Tangent Line:

   • Based on the graph, we can see that the tangent line at the rightmost point of the circle is at $x = 8$, but this is not one of the options.
   • Instead, we are asked for the tangent at $x = 6$, which suggests that $x = 6$ is very close to the tangent line on the right-hand side of the circle, although it does not coincide exactly with the rightmost point.

Conclusion:

The correct equation for the tangent line based on the given options is: $\boxed{x = 6}$

Let me know if you would like further clarification or have any questions.

Here are 5 related questions to further explore this topic:

1. What is the general form of a circle's equation?
2. How do you determine the slope of a tangent line to a circle?
3. What is the geometric definition of a tangent line?
4. How do you find the point of tangency for a given line and circle?
5. What changes if the circle’s center is shifted from the origin?

Tip: When dealing with circles, always check the distance from the center to a potential tangent line to verify tangency.