The equation of a circle has to be determined where the diameter is a line segment joining the points (3, 2) and (4, 6).

First, determine the center of the circle. This is the mid point of the line segment. The coordinates of the point are `((3 +4)/2, (2+6)/2) = (3.5, 4)`

The radius of the circle is half the distance between the two points. This is equal to `(1/2)*sqrt((4-3)^2+(6-2)^2)` = `(1/2)*sqrt(1+16)` = `(sqrt 17)/2`

The equation of a circle with radius `(sqrt 17)/2` and center (3.5, 4) is `(x - 3.5)^2 + (y-4)^2 = ((sqrt 17)/2)^2`

=> `(x - 3.5)^2 + (y-4)^2 = 17/4`

The equation of the required circle is `(x - 3.5)^2 + (y-4)^2 = 17/4`