Intuitively we're being asked how many points on the radius 1 circle centered at (4,1) are three units from the y axis. The answer is pretty obviously
Answer: One point
We'll work it out the usual way.
Let's give coordinates (x,y) to a point 3 units from the y axis and 1 unit from (4,1).
The x coordinate tells us how far we are from the y axis. 3 units from the y axis means
|x| = 3
x = ±3
One unit from (4,1) means
(x - 4)² + (y - 1)² = 1²
Let's try x=-3
(-7)² + (y - 1)² = 1
(y - 1)² = -48
No solution there
Let's try x=3
(3 - 4)² + (y - 1)² = 1
(-1)² + (y - 1)² = 1
(y - 1)² = 0
y - 1 = 0
y = 1
So (3,1) is the only point 3 units from the y axis and 1 unit from (4,1).
That verifies our answer above.