Key Concepts and Formulas

• Equation of a Circle Touching a Line at a Point: The equation of a circle touching the line $ax + by + c = 0$ at point $(x_1, y_1)$ is given by $(x - x_1)^2 + (y - y_1)^2 + \lambda(ax + by + c) = 0$, where $\lambda$ is a parameter.

• Equation of the Common Chord: The equation of the common chord of two circles $S_1 = 0$ and $S_2 = 0$ is given by $S_1 - S_2 = 0$.

• Circle Equation and Center: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, with center $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$.

Step-by-Step Solution

Step 1: Formulate the Equation of Circle C

• Why this step? We need to find the equation of circle C using the given tangency condition.
• The line is $x - 2y = 0$, and the point of tangency is $(2, 1)$.
• Using the formula for a circle touching a line at a point, we have: $(x - 2)^2 + (y - 1)^2 + \lambda(x - 2y) = 0$
• Expanding this, we get: $x^2 - 4x + 4 + y^2 - 2y + 1 + \lambda x - 2\lambda y = 0$ $x^2 + y^2 + (\lambda - 4)x + (-2 - 2\lambda)y + 5 = 0 \quad \text{(Equation of C)}$

Step 2: Identify the Center of Circle $C_1$

• Why this step? We need to find the center of circle $C_1$ because the common chord PQ is a diameter of $C_1$, meaning it passes through the center of $C_1$.
• The equation of circle $C_1$ is $x^2 + y^2 + 2y - 5 = 0$.
• Comparing with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = 0$, $2f = 2$, and $c = -5$. Thus, $g = 0$ and $f = 1$.
• The center of $C_1$ is $(-g, -f) = (0, -1)$.

Step 3: Determine the Equation of the Common Chord PQ

• Why this step? We need the equation of the common chord PQ to use the fact that it passes through the center of $C_1$.
• Let $S_C = x^2 + y^2 + (\lambda - 4)x + (-2 - 2\lambda)y + 5 = 0$.
• Let $S_{C_1} = x^2 + y^2 + 2y - 5 = 0$.
• The equation of the common chord PQ is $S_C - S_{C_1} = 0$: $(x^2 + y^2 + (\lambda - 4)x + (-2 - 2\lambda)y + 5) - (x^2 + y^2 + 2y - 5) = 0$ $(\lambda - 4)x + (-2 - 2\lambda - 2)y + 10 = 0$ $(\lambda - 4)x + (-4 - 2\lambda)y + 10 = 0 \quad \text{(Equation of PQ)}$

Step 4: Use the Diameter Condition to Find $\lambda$

• Why this step? We can substitute the coordinates of the center of $C_1$ into the equation of PQ to solve for $\lambda$.
• Substitute $(x, y) = (0, -1)$ into the equation of PQ: $(\lambda - 4)(0) + (-4 - 2\lambda)(-1) + 10 = 0$ $0 + 4 + 2\lambda + 10 = 0$ $2\lambda + 14 = 0$ $2\lambda = -14$ $\lambda = -7$

Step 5: Substitute $\lambda$ back into the Equation of Circle C

• Why this step? We can now find the specific equation of circle C by substituting the value of $\lambda$.
• Substitute $\lambda = -7$ into the equation of C: $x^2 + y^2 + (-7 - 4)x + (-2 - 2(-7))y + 5 = 0$ $x^2 + y^2 - 11x + (-2 + 14)y + 5 = 0$ $x^2 + y^2 - 11x + 12y + 5 = 0$

Step 6: Calculate the Diameter of Circle C

• Why this step? We need to find the diameter of circle C using its equation.
• The equation of circle C is $x^2 + y^2 - 11x + 12y + 5 = 0$.
• Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = -11$, $2f = 12$, and $c = 5$. Thus, $g = -\frac{11}{2}$ and $f = 6$.
• The radius $r$ of circle C is $\sqrt{g^2 + f^2 - c}$: $r = \sqrt{\left(-\frac{11}{2}\right)^2 + (6)^2 - 5}$ $r = \sqrt{\frac{121}{4} + 36 - 5}$ $r = \sqrt{\frac{121}{4} + 31}$ $r = \sqrt{\frac{121 + 124}{4}}$ $r = \sqrt{\frac{245}{4}}$ $r = \frac{\sqrt{245}}{2}$
• The diameter of circle C is $2r$: $\text{Diameter} = 2 \times \frac{\sqrt{245}}{2} = \sqrt{245} = \sqrt{49 \times 5} = 7\sqrt{5}$

Common Mistakes & Tips:

• Be careful with signs when calculating the center and radius of the circles.
• Remember to use the correct formula for the equation of a circle touching a line at a given point.
• Double-check algebraic manipulations to avoid errors.

Summary

We found the equation of circle C using the tangency condition and the common chord property. By finding the center of circle $C_1$ and using the fact that the common chord PQ is a diameter of $C_1$, we determined the value of $\lambda$. Substituting this value back into the equation of circle C, we found its specific equation and then calculated its diameter. The diameter of circle C is $7\sqrt{5}$.

The final answer is $\boxed{7\sqrt{5}}$, which corresponds to option (A).