Key Concepts and Formulas

• Radius is Perpendicular to Tangent: The radius drawn from the center of a circle to the point of tangency is always perpendicular to the tangent line at that point.
• Slopes of Perpendicular Lines: If two non-vertical lines are perpendicular, the product of their slopes is $-1$. If $m_1$ and $m_2$ are their respective slopes, then $m_1 \times m_2 = -1$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step-by-Step Solution

Step 1: Determine the Slope of the Tangent Line

The equation of the tangent line is given as $2x - y + 1 = 0$. To find its slope, we rewrite the equation in the slope-intercept form, $y = mx + c$. $y = 2x + 1$ Comparing this with $y = mx + c$, we identify the slope of the tangent line, $m_T$. $m_T = 2$ Explanation: Finding the slope of the tangent line is crucial because the radius at the point of tangency is perpendicular to it. Knowing the tangent's slope allows us to determine the slope of the radius using the perpendicularity condition.

Step 2: Express the Coordinates of the Circle's Center

Let the center of the circle be $C(h, k)$. We are given that the center lies on the line $x - 2y = 4$. Substituting the coordinates of the center into this equation: $h - 2k = 4$ From this, we can express $h$ in terms of $k$ or vice versa. Let's express $h$ in terms of $k$: $h = 2k + 4$ So, the coordinates of the center can be written as $C(2k + 4, k)$. Explanation: The condition that the center lies on a specific line helps us reduce the number of unknown variables for the center from two ($h$ and $k$) to just one ($k$). This simplifies our calculations in the following steps.

Step 3: Determine the Slope of the Radius

The radius connects the center $C(2k + 4, k)$ to the point of tangency $P(2, 5)$. Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, the slope of the radius, $m_R$, is: $m_R = \frac{5 - k}{2 - (2k + 4)}$ $m_R = \frac{5 - k}{2 - 2k - 4}$ $m_R = \frac{5 - k}{-2 - 2k}$ Explanation: We calculate the slope of the radius using the general coordinates of the center and the specific coordinates of the point of tangency. This slope will be used with the perpendicularity condition.

Step 4: Apply the Perpendicularity Condition to Find the Center

The radius is perpendicular to the tangent at the point of tangency. Therefore, the product of their slopes must be $-1$: $m_R \times m_T = -1$ Substitute the expressions for $m_R$ and $m_T$: $\left(\frac{5 - k}{-2 - 2k}\right) \times 2 = -1$ $\frac{5 - k}{-2 - 2k} = -\frac{1}{2}$ Now, we solve for $k$: $2(5 - k) = -1(-2 - 2k)$ $10 - 2k = 2 + 2k$ $8 = 4k$ $k = 2$ Now that we have $k$, we can find $h$ using the relation from Step 2: $h = 2k + 4 = 2(2) + 4 = 4 + 4 = 8$ So, the center of the circle is $C(8, 2)$. Explanation: This is the critical step where we use the geometric property to form an equation and solve for the unknown coordinates of the center. By finding $h$ and $k$, we have precisely located the center of the circle.

Step 5: Calculate the Radius of the Circle

The radius of the circle is the distance between the center $C(8, 2)$ and the point of tangency $P(2, 5)$. Using the distance formula $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$: $r = \sqrt{(8 - 2)^2 + (2 - 5)^2}$ $r = \sqrt{(6)^2 + (-3)^2}$ $r = \sqrt{36 + 9}$ $r = \sqrt{45}$ To simplify the radical, we find the largest perfect square factor of 45, which is 9 ($45 = 9 \times 5$): $r = \sqrt{9 \times 5}$ $r = \sqrt{9} \times \sqrt{5}$ $r = 3\sqrt{5}$ Explanation: Once the center and a point on the circle (the point of tangency) are known, the radius is simply the distance between these two points. This is the final step to answer the question.

Common Mistakes & Tips:

• Sign Errors: Be careful with positive and negative signs, especially when calculating slopes and manipulating algebraic expressions.
• Algebraic Simplification: Double-check your algebraic simplification steps, especially when dealing with fractions.
• Using the correct relationships: Ensure you have the correct relationship between the slopes of perpendicular lines.

Summary

This problem demonstrates how combining geometric properties (radius perpendicular to tangent) with algebraic techniques (slopes, solving linear equations, distance formula) allows us to determine the characteristics of a circle. The key is to systematically apply the known properties to set up equations and solve for the unknowns. The perpendicularity condition is a powerful tool when dealing with tangents to circles. The radius of the circle is $3\sqrt{5}$.

The final answer is $\boxed{3\sqrt{5}}$, which corresponds to option (C).