To find the equation of the circle, we need to follow these steps:

1. Find the equation of the line passing through the given points (3,7) and (5,5).
2. Find the equation of the perpendicular bisector of the line segment joining the two given points. This will give us the line on which the center of the circle lies.
3. Find the intersection point of the line x-4y=1 and the perpendicular bisector. This will give us the center of the circle.
4. Find the radius of the circle by calculating the distance between the center and any of the given points.
5. Use the center and radius to write the equation of the circle.

Let's start with step 1:

1. Finding the equation of the line passing through the points (3,7) and (5,5):

We can use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept.

The slope of the line can be found using the formula: m = (y2 - y1) / (x2 - x1)

Plugging in the values, we have:
m = (5 - 7) / (5 - 3)
m = -2 / 2
m = -1

Now, we can use the slope-intercept form:
y = -x + b

To find the value of b, we can substitute the coordinates of one of the given points into the equation:
7 = -(3) + b
b = 7 + 3
b = 10

Therefore, the equation of the line passing through the points (3,7) and (5,5) is:
y = -x + 10

Moving on to step 2:

2. Finding the equation of the perpendicular bisector:

The perpendicular bisector of a line segment is a line that intersects the segment at its midpoint and is perpendicular to it. The midpoint of the line segment joining the points (3,7) and (5,5) can be found using the midpoint formula:

Midpoint (x, y) = ((x1 + x2)/2, (y1 + y2)/2)

Plugging in the values, we have:
Midpoint (x, y) = ((3 + 5)/2, (7 + 5)/2)
Midpoint (x, y) = (4, 6)

Now, we can find the slope of the line joining the two given points using the formula: m = (y2 - y1) / (x2 - x1)

Plugging in the values, we have:
m = (5 - 7) / (5 - 3)
m = -2 / 2
m = -1

The slope of the perpendicular bisector is the negative reciprocal of the slope of the line joining the two points. Therefore, the slope of the perpendicular bisector is 1.

Using the point-slope form of a line, which is y - y1 = m(x - x1), we can write the equation of the perpendicular bisector passing through the midpoint (4, 6):

y - 6 = 1(x - 4)
y - 6 = x - 4
y = x + 2

Moving on to step 3:

3. Finding the intersection