1. Interpret the given condition

   The lines through $(3,2)$ parallel to the coordinate axes are:

   • vertical line: $x=3$
   • horizontal line: $y=2$

   These two lines are tangent to the circle $C$ of radius $1$.

2. Use the tangency condition to find the center

   Let the center of the circle be $(h,k)$ and radius be $1$.

   Since the line $x=3$ is tangent to the circle, the perpendicular distance from the center to this line must be $1$: $|h-3|=1$

   Since the line $y=2$ is tangent to the circle, the perpendicular distance from the center to this line must be $1$: $|k-2|=1$

   So, $h=3\pm 1 \,\Rightarrow\, h=2 \text{ or } 4$ $k=2\pm 1 \,\Rightarrow\, k=1 \text{ or } 3$

   Hence possible centers are: $(2,1),\ (2,3),\ (4,1),\ (4,3)$

3. Use the condition “closer to the origin”

   Compute distance of each possible center from the origin:

   • For $(2,1)$: $\sqrt{2^2+1^2}=\sqrt5$
   • For $(2,3)$: $\sqrt{2^2+3^2}=\sqrt{13}$
   • For $(4,1)$: $\sqrt{4^2+1^2}=\sqrt{17}$
   • For $(4,3)$: $\sqrt{4^2+3^2}=5$

   The circle closer to the origin has center $(2,1)$.

4. Find distance from $(5,5)$ to the center

   Distance between $(5,5)$ and center $(2,1)$ is: $d=\sqrt{(5-2)^2+(5-1)^2}$ $d=\sqrt{3^2+4^2}=\sqrt{25}=5$

5. Find the shortest distance from the point to the circle

   Since the radius is $1$, the shortest distance from the point to the circle is: $d-r=5-1=4$

6. Check options

   • A: $4\sqrt2$ ❌
   • B: $4$ ✅
   • C: $5$ ❌
   • D: $2\sqrt2$ ❌

Therefore, the correct answer is B.