Hello, realintegerz!
1) Find an eq. for a circle that has its center on the yaxis, that has a radius of 4,
and that has only one point on it whose xcoordinate is the same as its ycoordinate Code:

* * *
*  *
*  *
*  *

*  *
* (0,k)* *
*  *

*  *
*  *
*  o P
* * *


       +       

The circle has center (0,k) and radius 4.
It equation is: . .[1]
There is one point where the xcoordinate and ycoordinate are equal.
. .
Substitute into [1]: .
The Quadratic Formula gives us: .
It would seem that there are two such points.
Since we are told there is only one such point, the discriminant must be zero.
. . That is: .
Hence: .
There are two possible circles:
. .