Key Concepts and Formulas

• Equation of a Circle: A circle with center $(h, k)$ and radius $r$ has the equation $(x - h)^2 + (y - k)^2 = r^2$.
• Tangents Parallel to Axes: If the lines $x = a$ and $y = b$ are tangent to a circle, then the distance from the center of the circle to each line must equal the radius. Therefore, $|h - a| = r$ and $|k - b| = r$.
• Distance Formula: The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step-by-Step Solution

Step 1: Determine the equations of the tangent lines. Since the lines pass through the point $(3, 2)$ and are parallel to the coordinate axes, their equations are $x = 3$ and $y = 2$.

Step 2: Determine the center of the circle. Let the center of the circle be $(h, k)$. Since the lines $x = 3$ and $y = 2$ are tangent to the circle and the radius is 1, we have $|h - 3| = 1$ and $|k - 2| = 1$. This gives us four possible centers: $(4, 3), (4, 1), (2, 3), (2, 1)$. Since the circle is closer to the origin, we choose the center with smaller coordinates. Thus, the center is $(2, 1)$.

Step 3: Write the equation of the circle. The equation of the circle with center $(2, 1)$ and radius 1 is $(x - 2)^2 + (y - 1)^2 = 1$.

Step 4: Find the distance between the point $(5, 5)$ and the center of the circle. The distance between the point $(5, 5)$ and the center $(2, 1)$ is $d = \sqrt{(5 - 2)^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Step 5: Calculate the shortest distance between the point and the circle. The shortest distance between the point $(5, 5)$ and the circle is the distance between the point and the center minus the radius of the circle. Shortest distance = $5 - 1 = 4$.

Step 6: Re-examine the possible circle centers, to ensure we selected the closest. The possible centers are $(4, 3), (4, 1), (2, 3), (2, 1)$. The problem states the circle is closer to the origin. We can compare the distance of each center to the origin. $d_1 = \sqrt{4^2 + 3^2} = 5$ $d_2 = \sqrt{4^2 + 1^2} = \sqrt{17}$ $d_3 = \sqrt{2^2 + 3^2} = \sqrt{13}$ $d_4 = \sqrt{2^2 + 1^2} = \sqrt{5}$ Since $\sqrt{5}$ is the smallest distance, the center $(2, 1)$ is indeed closest to the origin. The shortest distance from (5,5) to the circle is still 4. However, we are given that the correct answer is $4\sqrt{2}$, so there must be an error in the logic.

Let's reconsider the condition "lines passing through the point (3,2) and parallel to the coordinate axes touch it". This means x=3 and y=2 are tangent to the circle. Let the center be (h,k) and radius r. Then |h-3| = r and |k-2| = r. So h = 3 $\pm$ r and k = 2 $\pm$ r. The equation of the circle is $(x-h)^2 + (y-k)^2 = r^2$. We are given that r=1. So h = 3 $\pm$ 1, k = 2 $\pm$ 1, thus h can be 4 or 2 and k can be 3 or 1. Since the circle is closer to the origin, we consider (2,1) as the center. The distance to (5,5) is $\sqrt{(5-2)^2 + (5-1)^2} = \sqrt{9+16} = 5$. Subtracting the radius, we get 5-1 = 4. However, the correct answer is $4\sqrt{2}$. This suggests the radius is not 1. Let's reconsider the question. The radius is 1. The center is (h,k). |h-3| = 1 and |k-2| = 1. h = 3 $\pm$ 1 and k = 2 $\pm$ 1. So h can be 2 or 4 and k can be 1 or 3. The circle is closer to the origin. Centers are (2,1), (2,3), (4,1), (4,3). Their distances from the origin are $\sqrt{5}, \sqrt{13}, \sqrt{17}, 5$. So (2,1) is the closest. The distance from (5,5) to (2,1) is 5. Shortest distance is 5-1 = 4.

The provided answer is incorrect. Let's re-examine the problem statement. The lines passing through (3,2) and parallel to the axes touch the circle. So x=3 and y=2 are tangent. Let the center be (h,k) and radius r=1. So |h-3| = 1 and |k-2| = 1. Therefore, h = 3 $\pm$ 1 and k = 2 $\pm$ 1. So the possible centers are (2,1), (2,3), (4,1), (4,3). We want the circle closest to the origin. The distances from the origin are $\sqrt{5}, \sqrt{13}, \sqrt{17}, 5$. So (2,1) is closest. The distance from (5,5) to (2,1) is $\sqrt{(5-2)^2 + (5-1)^2} = \sqrt{9+16} = 5$. Since the radius is 1, the shortest distance is 5-1 = 4.

If we assume the radius is not 1 and try to make the answer $4\sqrt{2}$, then the distance from (5,5) to (2,1) minus r must equal $4\sqrt{2}$. So 5-r = $4\sqrt{2}$, which means r = 5 - $4\sqrt{2}$. Then |h-3| = 5 - $4\sqrt{2}$ and |k-2| = 5 - $4\sqrt{2}$. But the radius is given as 1.

The distance from (5,5) to the circle with center (2,1) and radius 1 is 4. Let's test center (2,3). The distance is $\sqrt{(5-2)^2 + (5-3)^2} = \sqrt{9+4} = \sqrt{13}$. The shortest distance is $\sqrt{13} - 1 \approx 2.6$. Let's test center (4,1). The distance is $\sqrt{(5-4)^2 + (5-1)^2} = \sqrt{1+16} = \sqrt{17}$. The shortest distance is $\sqrt{17} - 1 \approx 3.12$. Let's test center (4,3). The distance is $\sqrt{(5-4)^2 + (5-3)^2} = \sqrt{1+4} = \sqrt{5}$. The shortest distance is $\sqrt{5} - 1 \approx 1.24$. The center (4,3) is closer to (5,5) than (2,1), but further from the origin.

Common Mistakes & Tips

• Double-check the given information. In this case, the question states the circle is closer to the origin and has a radius of 1.
• Draw a diagram to visualize the problem. This can help you understand the relationships between the points, lines, and circles.
• Be careful with absolute values. Remember that $|x| = a$ means $x = a$ or $x = -a$.

Summary The lines $x=3$ and $y=2$ are tangent to the circle, and the circle has a radius of 1. This means the center of the circle is $(2, 1)$. The distance from the center of the circle to the point $(5, 5)$ is 5. The shortest distance from the point to the circle is the distance from the point to the center minus the radius, which is $5 - 1 = 4$. The correct answer is 4. Given that the correct answer is $4\sqrt{2}$, there is likely an error in the problem statement or the provided answer. Based on the problem as stated, the answer is 4.

The final answer is \boxed{4}. Option (B) is correct.