Introduction:
To find the locus of the middle point of the portion of the line intercepted between the axes, we need to determine the equation of the line passing through the point (3, -2) and then find the coordinates of the midpoint of the line segment intercepted between the axes.

Step 1: Finding the equation of the line:
Given that the line passes through the point (3, -2), we can use the point-slope form of a linear equation to determine the equation of the line. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.

Step 2: Calculating the slope:
To calculate the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
We can choose any other point on the line to calculate the slope. Let's choose the point (0, b) on the y-axis, where b is the y-intercept.

Step 3: Determining the equation of the line:
Using the slope formula and the point (3, -2), we have:
m = (-2 - b) / (3 - 0)

Simplifying the equation, we get:
m = (-2 - b) / 3

Substituting the value of m in the point-slope form, we get:
y - (-2) = ((-2 - b) / 3)(x - 3)

Simplifying further, we have:
y + 2 = (-2 - b)x/3 + 2b/3

Multiplying through by 3 to remove the fraction, we get:
3y + 6 = (-2 - b)x + 2b

Rearranging the terms, we finally have the equation of the line:
(-2 - b)x + 3y + (2b - 6) = 0

Step 4: Finding the coordinates of the midpoint:
To find the coordinates of the midpoint, we need to find the average of the x-coordinates and the average of the y-coordinates of the two intercepts with the axes.

The x-coordinate of the midpoint is given by:
x = (0 + 3) / 2 = 3/2 = 1.5

The y-coordinate of the midpoint is given by:
y = (b - 2 + 0) / 2 = (b - 2) / 2

Therefore, the locus of the midpoint is given by the equation:
y = (b - 2) / 2, where b is a constant.

Conclusion:
The locus of the midpoint of the portion of the line intercepted between the axes is given by the equation y = (b - 2) / 2, where b is a constant.

Answer