Given:
Points A (3,2) and B (-1,3)
Point P on y-axis equidistant from A and B

To find:
Coordinates of point P

Approach:
1. Find the distance between points A and B.
2. Find the midpoint of line AB.
3. Find the equation of the line passing through the midpoint and perpendicular to AB.
4. Find the intersection point of the line with the y-axis.

Calculation:
Step 1: Find the distance between points A and B using the distance formula.
Distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using the formula, we have:
dAB = sqrt((-1 - 3)^2 + (3 - 2)^2)
= sqrt((-4)^2 + (1)^2)
= sqrt(16 + 1)
= sqrt(17)

Step 2: Find the midpoint of line AB.
Midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Using the formula, we have:
Midpoint = ((-1 + 3)/2, (3 + 2)/2)
= (2/2, 5/2)
= (1, 5/2)

Step 3: Find the equation of the line passing through the midpoint and perpendicular to AB.
The slope of line AB is given by:
mAB = (y2 - y1)/(x2 - x1)

Using the formula, we have:
mAB = (3 - 2)/(-1 - 3)
= 1/(-4)
= -1/4

The slope of the line perpendicular to AB is the negative reciprocal of mAB.
m_perpendicular = -1/mAB
= -1/(-1/4)
= 4

Using the point-slope form of a line, the equation of the line passing through the midpoint is:
y - y1 = m(x - x1)

Substituting the values, we have:
y - (5/2) = 4(x - 1)

Simplifying the equation, we get:
y - 5/2 = 4x - 4

Step 4: Find the intersection point of the line with the y-axis.
To find the intersection point, substitute x = 0 in the equation of the line:
y - 5/2 = 4(0) - 4
y - 5/2 = -4

Simplifying the equation, we have:
y = -4 + 5/2
y = -8/2 + 5/2
y = -3/2

Therefore, the coordinates of point P are (0, -3/2).

Hence, the correct answer is option B) (0, -3/2).