Two circles of radius 1 are centered at $(4,0)$ and $(-4,0).$ How many circles are tangent to both of the given circles and also pass through the point $(0,5)$?

1 circle

Step-by-step explanation:

Given two circles (red circles on the diagram).

There are two tangent circles to both of the given circles (blue circles on the diagram), and only one of them is passing through the point (0,5).

Let's check it.

The equations of the tangent circles are

Check whether point (0,5) lies on the smaller circle:

No

Check whether point (0,5) lies on the larger circle:

Yes

1 circle