Key Concepts and Formulas

• The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$. The general form is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$.
• Rationalizing the denominator: $\frac{1}{a - b\sqrt{c}} = \frac{a + b\sqrt{c}}{(a - b\sqrt{c})(a + b\sqrt{c})} = \frac{a + b\sqrt{c}}{a^2 - b^2c}$.

Step-by-Step Solution

Step 1: Understand the Problem and Identify Given Information

We are given a fixed point $P(-4, 1)$ and a circle whose equation is $x^2 + y^2 + 2x + 4y - 4 = 0$. We need to find the radii of the largest ($r_1$) and smallest ($r_2$) circles that pass through $P$ and have their centers on the given circle. Finally, we need to find $a + b$, where $\frac{r_1}{r_2} = a + b\sqrt{2}$.

Step 2: Find the Center and Radius of the Locus Circle

The equation of the given circle is $x^2 + y^2 + 2x + 4y - 4 = 0$. Comparing this to the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = 2$, $2f = 4$, and $c = -4$. Thus, $g = 1$, $f = 2$, and $c = -4$.

The center of the circle, $C_L$, is $(-g, -f) = (-1, -2)$. The radius of the circle, $R_L$, is $\sqrt{g^2 + f^2 - c} = \sqrt{1^2 + 2^2 - (-4)} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$.

So, the locus circle has center $C_L(-1, -2)$ and radius $R_L = 3$.

Step 3: Calculate the Distance Between the Fixed Point $P$ and the Center of the Locus Circle $C_L$

The fixed point is $P(-4, 1)$ and the center of the locus circle is $C_L(-1, -2)$. Using the distance formula, the distance $PC_L$ is: $PC_L = \sqrt{(-1 - (-4))^2 + (-2 - 1)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$

Step 4: Determine the Maximum and Minimum Radii $r_1$ and $r_2$

The maximum radius $r_1$ is the distance from $P$ to $C_L$ plus the radius of the locus circle, $R_L$. $r_1 = PC_L + R_L = 3\sqrt{2} + 3$

The minimum radius $r_2$ is the absolute difference between the distance from $P$ to $C_L$ and the radius of the locus circle, $R_L$. Since $PC_L > R_L$, $r_2 = PC_L - R_L = 3\sqrt{2} - 3$

Step 5: Calculate the Ratio $\frac{r_1}{r_2}$

$\frac{r_1}{r_2} = \frac{3\sqrt{2} + 3}{3\sqrt{2} - 3} = \frac{\sqrt{2} + 1}{\sqrt{2} - 1}$ To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{2} + 1$: $\frac{\sqrt{2} + 1}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{(\sqrt{2} + 1)^2}{(\sqrt{2})^2 - (1)^2} = \frac{2 + 2\sqrt{2} + 1}{2 - 1} = \frac{3 + 2\sqrt{2}}{1} = 3 + 2\sqrt{2}$

Step 6: Find $a + b$

We have $\frac{r_1}{r_2} = 3 + 2\sqrt{2}$, so $a = 3$ and $b = 2$. Therefore, $a + b = 3 + 2 = 5$.

Common Mistakes & Tips

• Always remember to rationalize the denominator when simplifying expressions involving radicals.
• Be careful when calculating the minimum radius. If the point $P$ were inside the locus circle, the minimum radius would be $R_L - PC_L$. Use $|PC_L - R_L|$.
• Ensure you correctly identify the center and radius of the locus circle from its equation.

Summary

We found the center and radius of the locus circle, calculated the distance between the fixed point and the center of the locus circle, and then determined the maximum and minimum radii of the circles passing through the fixed point and having their centers on the locus circle. Finally, we calculated the ratio of the maximum to minimum radius, rationalized the denominator, and found the sum of the coefficients $a$ and $b$.

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