Key Concepts and Formulas

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Equation of a Circle: A circle with center $(h, k)$ and radius $r$ has the equation $(x - h)^2 + (y - k)^2 = r^2$.
• External Tangency of Circles: Two circles with centers $C_1$ and $C_2$ and radii $r_1$ and $r_2$ touch externally if the distance between their centers is equal to the sum of their radii: $Distance(C_1, C_2) = r_1 + r_2$.

Step 1: Analyze the given Circle C

We are given that circle $C$ has center at $(1, 1)$ and radius $1$. Let $C_C$ denote the center of circle $C$ and $R_C$ its radius. Thus, $C_C = (1, 1)$ and $R_C = 1$.

Step 2: Determine the properties of Circle T

Circle $T$ is centered at $(0, y)$ and passes through the origin $(0, 0)$. Let $C_T$ denote the center of circle $T$ and $R_T$ its radius. Thus, $C_T = (0, y)$. Since the circle passes through the origin, the radius $R_T$ is the distance between $(0, y)$ and $(0, 0)$. $R_T = \sqrt{(0 - 0)^2 + (y - 0)^2} = \sqrt{y^2} = |y|$ Since $R_T$ represents the radius of a circle, it must be positive. Thus, $R_T = |y|$.

Step 3: Apply the condition for external tangency

Circle $T$ touches circle $C$ externally. Therefore, the distance between their centers equals the sum of their radii: $Distance(C_C, C_T) = R_C + R_T$ We have $C_C = (1, 1)$, $C_T = (0, y)$, $R_C = 1$, and $R_T = |y|$. The distance between the centers is: $Distance(C_C, C_T) = \sqrt{(1 - 0)^2 + (1 - y)^2} = \sqrt{1 + (1 - y)^2}$ Thus, $\sqrt{1 + (1 - y)^2} = 1 + |y|$

Step 4: Solve the equation for y

Square both sides of the equation to eliminate the square root: $1 + (1 - y)^2 = (1 + |y|)^2$ $1 + (1 - 2y + y^2) = 1 + 2|y| + y^2$ $2 - 2y + y^2 = 1 + 2|y| + y^2$ Subtract $y^2$ from both sides: $2 - 2y = 1 + 2|y|$ $1 - 2y = 2|y|$

Step 5: Handle the absolute value

We consider two cases:

Case 1: $y \ge 0$. Then $|y| = y$. $1 - 2y = 2y$ $1 = 4y$ $y = \frac{1}{4}$ Since $y = \frac{1}{4} \ge 0$, this solution is valid.

Case 2: $y < 0$. Then $|y| = -y$. $1 - 2y = -2y$ $1 = 0$ This is a contradiction, so there are no solutions for $y < 0$.

Therefore, the only valid solution is $y = \frac{1}{4}$.

Step 6: Determine the radius of Circle T

Since $y = \frac{1}{4}$, the radius of circle $T$ is: $R_T = |y| = \left|\frac{1}{4}\right| = \frac{1}{4}$

Common Mistakes & Tips

• Remember that $\sqrt{y^2} = |y|$, not just $y$. Always consider the absolute value when taking the square root of a squared variable.
• When squaring equations, check for extraneous solutions. In this case, the negative case was extraneous.
• Drawing a diagram can help visualize the problem and understand the geometric relationships.

Summary

By using the condition for external tangency and carefully handling the absolute value, we found that the $y$-coordinate of the center of circle $T$ is $\frac{1}{4}$. This means the radius of circle $T$ is $\frac{1}{4}$.

Final Answer

The final answer is $\boxed{1/4}$, which corresponds to option (B).