Key Concepts and Formulas

• Section Formula: If point P$(x, y)$ divides the line segment joining A$(x_1, y_1)$ and B$(x_2, y_2)$ in the ratio $m:n$, then $x = \frac{mx_2 + nx_1}{m+n}$ and $y = \frac{my_2 + ny_1}{m+n}$.
• Equation of a Circle: The standard equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.
• Distance Formula: The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Parametric Representation of a Circle: A point on the circle $x^2 + y^2 = r^2$ can be represented as $(r\cos\theta, r\sin\theta)$.

Step-by-Step Solution

Step 1: Define the points and the given ratio.

We are given point A$(1, 2)$, and point B lies on the circle $x^2 + y^2 = 16$. Point P divides AB in the ratio $3:2$. We want to find the length of AC, where C is the center of the locus of P.

Step 2: Represent point B parametrically.

Since B lies on the circle $x^2 + y^2 = 16$, which has a radius of 4, we can represent B as $(4\cos\theta, 4\sin\theta)$.

Step 3: Apply the section formula to find the coordinates of P.

Let P be $(h, k)$. Since P divides AB in the ratio $3:2$, we can use the section formula: $h = \frac{3(4\cos\theta) + 2(1)}{3+2} = \frac{12\cos\theta + 2}{5}$ $k = \frac{3(4\sin\theta) + 2(2)}{3+2} = \frac{12\sin\theta + 4}{5}$

Step 4: Eliminate $\theta$ to find the locus of P.

From the equations in Step 3, we can isolate $\cos\theta$ and $\sin\theta$: $5h = 12\cos\theta + 2 \implies \cos\theta = \frac{5h - 2}{12}$ $5k = 12\sin\theta + 4 \implies \sin\theta = \frac{5k - 4}{12}$ Now, use the trigonometric identity $\cos^2\theta + \sin^2\theta = 1$: $\left(\frac{5h - 2}{12}\right)^2 + \left(\frac{5k - 4}{12}\right)^2 = 1$ $(5h - 2)^2 + (5k - 4)^2 = 144$

Step 5: Rewrite the equation in the standard form of a circle.

Expand and rearrange the equation: $25h^2 - 20h + 4 + 25k^2 - 40k + 16 = 144$ $25h^2 + 25k^2 - 20h - 40k = 124$ Divide by 25: $h^2 + k^2 - \frac{4}{5}h - \frac{8}{5}k = \frac{124}{25}$ Complete the square: $\left(h - \frac{2}{5}\right)^2 + \left(k - \frac{4}{5}\right)^2 = \frac{124}{25} + \frac{4}{25} + \frac{16}{25} = \frac{144}{25}$ Replacing $(h,k)$ with $(x,y)$: $\left(x - \frac{2}{5}\right)^2 + \left(y - \frac{4}{5}\right)^2 = \frac{144}{25}$ This is the equation of a circle with center $C\left(\frac{2}{5}, \frac{4}{5}\right)$.

Step 6: Calculate the length of AC.

We have $A(1, 2)$ and $C\left(\frac{2}{5}, \frac{4}{5}\right)$. Using the distance formula: $AC = \sqrt{\left(\frac{2}{5} - 1\right)^2 + \left(\frac{4}{5} - 2\right)^2} = \sqrt{\left(-\frac{3}{5}\right)^2 + \left(-\frac{6}{5}\right)^2} = \sqrt{\frac{9}{25} + \frac{36}{25}} = \sqrt{\frac{45}{25}} = \sqrt{\frac{9 \cdot 5}{25}} = \frac{3\sqrt{5}}{5}$

Common Mistakes & Tips

• When using the section formula, make sure you are using the correct ratio and assigning the correct coordinates to A and B.
• Be careful with algebraic manipulations, especially when completing the square or simplifying fractions.
• Remember the parametric form of the circle: $x = r\cos\theta$ and $y = r\sin\theta$.

Summary

We found the locus of point P, which divides the line segment AB in the ratio 3:2, where A is (1, 2) and B lies on the circle $x^2 + y^2 = 16$. By using the section formula and eliminating the parameter $\theta$, we found the center of the locus of P to be $(\frac{2}{5}, \frac{4}{5})$. Finally, we calculated the distance between A and the center C, which is $\frac{3\sqrt{5}}{5}$.

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