Key Concepts and Formulas

• Equation of Tangent to a Circle: The equation of the tangent to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ at the point $(x_1, y_1)$ is $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$. For a circle $x^2 + y^2 = r^2$, the tangent at $(x_1, y_1)$ is $xx_1 + yy_1 = r^2$.
• Distance from a Point to a Line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.
• Relationship between Radius, Chord Length, and Distance: If $r$ is the radius of a circle, $l$ is the length of a chord, and $d$ is the perpendicular distance from the center of the circle to the chord, then $r^2 = (\frac{l}{2})^2 + d^2$.

Step-by-Step Solution

Step 1: Finding the Equation of the Tangent to Circle $C_1$

• Objective: Determine the equation of the tangent to the circle $C_1: x^2 + y^2 - 2x - 1 = 0$ at the point $(2, 1)$. This tangent will also be the chord of circle $C_2$.
• Given Information:
  • Equation of circle $C_1$: $x^2 + y^2 - 2x - 1 = 0$
  • Point of tangency: $(2, 1)$
• Procedure:
  1. Rewrite the equation of $C_1$ as $x^2 + y^2 + 2gx + 2fy + c = 0$. Comparing with $x^2 + y^2 - 2x - 1 = 0$, we have $g = -1$, $f = 0$, and $c = -1$.
  2. Apply the tangent equation formula: $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$.
  3. Substitute the values: $x(2) + y(1) + (-1)(x+2) + (0)(y+1) + (-1) = 0$.
  4. Simplify: $2x + y - x - 2 - 1 = 0$, which simplifies to $x + y - 3 = 0$ or $x + y = 3$.
• Explanation: This step finds the equation of the tangent line to $C_1$ at $(2, 1)$. This line is also the chord of $C_2$.

Step 2: Identifying the Chord of Circle $C_2$

• Objective: Clearly define the equation representing the chord of circle $C_2$.
• Procedure: As determined in Step 1, the tangent to $C_1$ serves as the chord of $C_2$.
• Result: The equation of the chord for $C_2$ is $x + y - 3 = 0$.
• Explanation: This step establishes the connection between the circles $C_1$ and $C_2$ through their shared chord.

Step 3: Calculating the Perpendicular Distance from the Center of $C_2$ to the Chord

• Objective: Calculate the perpendicular distance ($d$) from the center of $C_2$, which is $(3, -2)$, to the chord $x + y - 3 = 0$.
• Given Information:
  • Center of $C_2$: $(3, -2)$
  • Equation of the chord: $x + y - 3 = 0$
• Procedure:
  1. Apply the perpendicular distance formula: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.
  2. Substitute the values: $d = \frac{|(1)(3) + (1)(-2) - 3|}{\sqrt{1^2 + 1^2}}$.
  3. Simplify: $d = \frac{|3 - 2 - 3|}{\sqrt{2}} = \frac{|-2|}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$.
• Explanation: This step determines the perpendicular distance from the center of $C_2$ to the chord, which is required to calculate the radius of $C_2$.

Step 4: Finding the Radius of Circle $C_2$

• Objective: Calculate the radius ($r$) of circle $C_2$ using the chord length and the perpendicular distance.

• Given Information:

  • Chord length: $l = 4$
  • Perpendicular distance: $d = \sqrt{2}$

• Procedure:

  1. Use the relationship $r^2 = (\frac{l}{2})^2 + d^2$.
  2. Substitute the values: $r^2 = (\frac{4}{2})^2 + (\sqrt{2})^2$.
  3. Simplify: $r^2 = (2)^2 + 2 = 4 + 2 = 6$.
  4. Therefore, $r = \sqrt{6}$.

• Explanation: This step uses the Pythagorean theorem relationship to find the radius of circle $C_2$ using the chord length and the distance from the center to the chord.

Step 5: Correcting the Error There was an error in the original solution. It should be $r = \sqrt{6}$, not 2.

Common Mistakes & Tips

• Sign Errors: Double-check the signs when substituting values into formulas, especially when dealing with negative coordinates or coefficients.
• Formula Confusion: Ensure you are using the correct formula for the equation of a tangent and the distance from a point to a line.
• Units: While not explicitly relevant here, always pay attention to units in geometric problems to avoid errors.

Summary

We first found the equation of the tangent to circle $C_1$ at the given point, which also serves as the chord of circle $C_2$. Then, we calculated the perpendicular distance from the center of $C_2$ to this chord. Finally, using the relationship between the radius, chord length, and perpendicular distance, we found the radius of $C_2$ to be $\sqrt{6}$.

Final Answer

The final answer is $\boxed{\sqrt{6}}$, which corresponds to option (D).