Given the points A(a,3) B (4, a+3)

Also, given that the distance between A and B is 4 units.

We will use the distance formula to solve:

We know that:

D = sqrt[ ( xB-xA)^2 + ( yB-yA)^2 ]

==> D(AB) = sqrt [ ( 4-a)^2 + ( a+3-3)^2 ] = 4

==> sqrt( 16 - 8a + a^2 + a^2) = 4

==> sqrt( 2a^2 - 8a + 16) = 4

Square both sides:

==> 2a^2 - 8a + 16) = 16

==> 2a^2 - 8a = 0

==> 2a( a - 4) = 0

==> a= 0 and a= 4

Then there are 2 possible solutions:

**a = { 0 , 4}**

We have the point A ( a,3) and B( 4, a+3) and the distance between them is 4 units. We have to find a.

Now the distance between 2 points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2]

Here we have

sqrt[(4 - a)^2 + (a +3 - 3)^2] = 4

square both the sides.

=> (4 - a)^2 + (a +3 - 3)^2 = 16

=> 16 + a^2 - 8a + a^2 = 16

=> 2a^2 - 8a = 0

=> a(2a - 8) = 0

=> a = 4 and a = 0

**Therefore the values of a can be 4 and 0.**