The only information missing is the circle's center; let it be the point (a, b). Then the equation of the circle is
(x - a)² + (y - b)² = 5²
We know the point (4, 1) lies on the circle, so we have the condition
(4 - a)² + (1 - b)² = 5² ⇒ (a - 4)² + (b - 1)² = 25
The circle just touches the line, so the line is tangent to the circle. The slope of the line is -3/4, since
3x + 4y - 16 = 0 ⇒ y = 4 - 3/4 x
Any non-vertical tangent line to the circle at (a', b') has slope equal to dy/dx with x = a' and y = b'. Differentiating both sides of the circle equation with respect to x yields
2 (x - a) + 2 (y - b) dy/dx = 0 ⇒ dy/dx = -(x - a)/(y - b)
At the point (4, 1), the slope of the tangent is -3/4, so we get the condition
-(4 - a)/(1 - b) = -3/4 ⇒ 4a - 3b = 13
Solve for b in terms of a and substitute into the other condition to solve for a :
4a - 3b = 13 ⇒ b = (4a - 13)/3
(a - 4)² + ((4a - 13)/3 - 1)² = 25
25/9 a² - 200/9 a + 400/9 = 25
25/9 a² - 200/9 a + 175/9 = 0
a² - 8a + 7 = 0
(a - 7) (a - 1) = 0
a = 7 or a = 1
Solve for b :
b = (4a - 13)/3 ⇒ b = 5 or b = -3
So there are two possible circles,
(x - 7)² + (y - 5)² = 5²
or
(x - 1)² + (y + 3)² = 5²