To find the equation of the circle and the coordinates of point $C$, we can follow these steps:

1. Equation of the Circle:

We know:

• The radius $r$ of the circle is given as 20.
• The center of the circle, $M$, is at $(29, 0)$.

The general form of the equation of a circle is: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

Using the given information:

• Center $M(29, 0)$ means $h = 29$ and $k = 0$.
• Radius $r = 20$.

The equation becomes: $(x - 29)^2 + y^2 = 20^2$ $(x - 29)^2 + y^2 = 400$ This is the equation of the circle.

2. Finding Point $C$:

From the problem, point $C$ is the intersection of the circle with the y-axis. On the y-axis, the x-coordinate is 0, so to find point $C$, substitute $x = 0$ into the equation of the circle.

The equation of the circle is: $(0 - 29)^2 + y^2 = 400$ $841 + y^2 = 400$ $y^2 = 400 - 841$ $y^2 = -441$ This results in a negative number under the square root, which implies that there is no real intersection of the circle with the y-axis in this case. Therefore, point $C$ as described doesn't exist with the given values, but this could suggest a check on the values or configuration of the system.

Would you like me to check this step further or assist with a different approach?

Additional Questions:

1. Would you like more clarification on why point $C$ might not exist with these parameters?
2. Should we review the given values for any inconsistencies?
3. Would you like to explore alternative methods for finding point $C$?
4. Should we verify point $B$ and $M$ coordinates before proceeding?
5. Are there any other details in the image you'd like to examine?

Tip:

Always double-check values or parameters like center and radius when results seem unexpected (like a negative value under the square root).