Key Concepts and Formulas

• Distance of a Point from a Line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by the formula: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$
• Condition for Tangency: A line is tangent to a circle if and only if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle.
• Side of a Line: For a line $Ax + By + C = 0$, two points $(x_1, y_1)$ and $(x_2, y_2)$ lie on the same side of the line if the expressions $Ax_1 + By_1 + C$ and $Ax_2 + By_2 + C$ have the same sign. They lie on opposite sides if these expressions have opposite signs.

Step-by-Step Solution

Step 1: Identify the centers and radii of the circles.

We are given three circles:

• $C_1: x^2 + y^2 = r^2$ has center $O_1 = (0, 0)$ and radius $r$.
• $C_2: (x - 1)^2 + (y - 1)^2 = r^2$ has center $O_2 = (1, 1)$ and radius $r$.
• $C_3: (x - 2)^2 + (y - 1)^2 = r^2$ has center $O_3 = (2, 1)$ and radius $r$.

This step is simply extracting the given information and writing it down in a clear format.

Step 2: Express the line equation in the general form.

The line $L: y = mx + c$ can be rewritten as $mx - y + c = 0$. This is needed to apply the distance formula effectively.

Step 3: Apply the tangency condition to each circle.

Since $L$ is tangent to all three circles, the distance from each center to the line must equal $r$.

• For $C_1$: $r = \frac{|m(0) - (0) + c|}{\sqrt{m^2 + (-1)^2}} = \frac{|c|}{\sqrt{m^2 + 1}}$
• For $C_2$: $r = \frac{|m(1) - (1) + c|}{\sqrt{m^2 + (-1)^2}} = \frac{|m - 1 + c|}{\sqrt{m^2 + 1}}$
• For $C_3$: $r = \frac{|m(2) - (1) + c|}{\sqrt{m^2 + (-1)^2}} = \frac{|2m - 1 + c|}{\sqrt{m^2 + 1}}$

This step utilizes the distance formula and the tangency condition to set up three equations.

Step 4: Use the "side" condition to eliminate absolute values.

The problem states that $C_1$ and $C_3$ are on one side of $L$, while $C_2$ is on the other side. This means that the expressions $c$ and $2m - 1 + c$ have the same sign, while $m - 1 + c$ has the opposite sign. Therefore:

• $c = -(m - 1 + c) \implies c = -m + 1 - c \implies m + 2c = 1$
• $2m - 1 + c = - (m - 1 + c) \implies 2m - 1 + c = -m + 1 - c \implies 3m + 2c = 2$

This step is critical. The side condition allows us to remove the absolute values and obtain a system of linear equations.

Step 5: Solve the system of equations for $m$ and $c$.

We have the following system of equations: $m + 2c = 1$ $3m + 2c = 2$

Subtracting the first equation from the second, we get: $2m = 1 \implies m = \frac{1}{2}$

Substituting $m = \frac{1}{2}$ into the first equation: $\frac{1}{2} + 2c = 1 \implies 2c = \frac{1}{2} \implies c = \frac{1}{4}$

Therefore, $m = \frac{1}{2}$ and $c = \frac{1}{4}$.

Step 6: Calculate $r^2$ using the value of $c$ and $m$.

Using the equation $r = \frac{|c|}{\sqrt{m^2 + 1}}$, we have: $r = \frac{|\frac{1}{4}|}{\sqrt{(\frac{1}{2})^2 + 1}} = \frac{\frac{1}{4}}{\sqrt{\frac{1}{4} + 1}} = \frac{\frac{1}{4}}{\sqrt{\frac{5}{4}}} = \frac{\frac{1}{4}}{\frac{\sqrt{5}}{2}} = \frac{1}{2\sqrt{5}}$ Therefore, $r^2 = \frac{1}{20}$.

Step 7: Calculate $20(r^2 + c)$.

We have $r^2 = \frac{1}{20}$ and $c = \frac{1}{4}$. So, $20(r^2 + c) = 20(\frac{1}{20} + \frac{1}{4}) = 20(\frac{1}{20} + \frac{5}{20}) = 20(\frac{6}{20}) = 6$

Therefore, $20(r^2 + c) = 6$.

Common Mistakes & Tips

• Ignoring the "side" condition: This is the most common mistake. Failing to correctly interpret and apply this condition will lead to incorrect values for $m$ and $c$.
• Sign errors: Be very careful with signs when substituting and solving equations. A small sign error can throw off the entire solution.
• Algebraic manipulation: Double-check your algebraic manipulations, especially when simplifying expressions and solving for variables.

Summary

We used the tangency condition along with the distance formula to set up equations relating $r$, $m$, and $c$. The crucial step was to correctly interpret the "side" condition, which allowed us to eliminate the absolute values and solve for $m$ and $c$. We then found $r^2$ and computed the value of $20(r^2 + c)$, which resulted in 6.

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