Given the point P(0, a) is midpoint of A(2,5) and B(b, 3).

We need to determine a and b.

We will use the midpoint formula to determine a and b.

We know that the midpoint formula is:

XP = ( xA+ xB) / 2

==> 0 = ( 2+ b) /2

We will multiply by 2.

==> 2*0 = 2+ b

==> 0 + 2+ b

**==> b= -2.**

Now we will determine y coordinates.

==> yP = ( yA + yB) / 2

==> a = ( 5+ 3 ) /2

==> a= 8/ 2 = 4

**==> a= 4**

**Then the point P(0, 4) is midpoint A(2, 5) and B(-2, 3).**

The coordinates of the midpoint of a segment is the arithmetical mean of the correspondent coordinates of the endpoints.

2xP = xA + xB

2yP = yA + yB

We'll substitute the coordinates for A and B and we'll get:

2*0 = 2 + b

We'll use symmetric property:

b + 2 = 0

We'll subtract 2 both sides:

**b = -2**

2a = 5 + 3

2a = 8

We'll divide by 2:

**a = 4**

To find the mid point M of (x1,y1) and (x2,y2) we use the formula.

Mx = (x1+x2)/2.

My = (y1+y2)/2.

We have to find P(0, a) which is the mid point of A(0,5) and B(b,3).

We use the same formula to determine P(0, a) or the y coordinate Py = a and the x coordinate Bx = b.

Therefore Px = 0 = (Ax+By)/2 = (0+b)/2.

0 = (0+b)/2

2*0 = 0+b. So b = 0.

Py = a = (Ay +By)/2 = (5+3)/2 = 8/4 =4.

Therefore a = 4.

Therefore a = 4 and b = 0 .