Let's first find the center of the circle. The center of a circle is the point where the perpendicular bisectors of its diameters intersect.

The slope of the line 3x - 4y - 7 = 0 can be found by rearranging it to slope-intercept form:

3x - 4y - 7 = 0

-4y = -3x + 7

y = (3/4)x - 7/4

So the slope of this line is 3/4.

The slope of the line 2x - 3y - 5 = 0 can also be found in slope-intercept form:

2x - 3y - 5 = 0

-3y = -2x + 5

y = (2/3)x - 5/3

So the slope of this line is 2/3.

The midpoint of the line segment connecting any two points on a diameter of a circle is the center of the circle. So let's find the midpoint of the line segment connecting two points on these lines.

To find the midpoint, we need to find the coordinates of two points on each line. We can do this by setting x = 0 and solving for y, and setting y = 0 and solving for x.

For the first line, when x = 0, we have:

3(0) - 4y - 7 = 0

y = -7/4

So one point on this line is (0, -7/4).

When y = 0, we have:

3x - 4(0) - 7 = 0

x = 7/3

So another point on this line is (7/3, 0).

For the second line, when x = 0, we have:

2(0) - 3y - 5 = 0

y = -5/3

So one point on this line is (0, -5/3).

When y = 0, we have:

2x - 3(0) - 5 = 0

x = 5/2

So another point on this line is (5/2, 0).

The midpoint of the line segment connecting (0, -7/4) and (7/3, 0) is:

((0 + 7/3)/2, (-7/4 + 0)/2) = (7/6, -7/8)

The midpoint of the line segment connecting (0, -5/3) and (5/2, 0) is:

((0 + 5/2)/2, (-5/3 + 0)/2) = (5/4, -5/6)

Now we need to find the point where the perpendicular bisectors of these two segments intersect.

The slope of the line connecting the two midpoints is:

(-7/8 - (-5/6))/(7/6 - 5/4) = -23/16

The negative reciprocal of this slope is the slope of a line perpendicular to the line connecting the midpoints. So the slope of the perpendicular bisector is:

16/23

We can use point-slope form to find the