To solve this problem, let's go through it step-by-step using the information provided.

Step 1: Set up the geometry and variables

1. Circle centers: We have two circles, one centered at $O$ with radius $r$ and the other centered at $P$ with radius $s$.

2. Intersection points: These circles intersect at points $A$ and $B$.

3. Key points on the line segment $OP$:

   • The line segment $OP$ has a length of $6$.
   • $OC = 2$, so $C$ is on the circle with center $O$.

   Since $OC = 2$ is on the circle centered at $O$ with radius $r$, it follows that $r = OC = 2$.

Step 2: Use cosine rule in $\triangle OAP$

We are given that $\angle AOP = 60^\circ$. Let $OP = 6$ and $OA = r = 2$. We aim to find the radius $s = PB$.

Applying the cosine rule in $\triangle OAP$: $AP^2 = OA^2 + OP^2 - 2 \cdot OA \cdot OP \cdot \cos(\angle AOP)$ Substitute the values: $AP^2 = 2^2 + 6^2 - 2 \cdot 2 \cdot 6 \cdot \cos(60^\circ)$ Since $\cos(60^\circ) = \frac{1}{2}$: $AP^2 = 4 + 36 - 2 \cdot 2 \cdot 6 \cdot \frac{1}{2}$ $AP^2 = 40 - 12 = 28$ Thus, $AP = \sqrt{28} = 2\sqrt{7}$

Step 3: Recognize $AP = s$

Since $A$ is on both circles, the distance $AP$ is the radius $s$ of the circle centered at $P$. Therefore: $s = 2\sqrt{7}$

Step 4: Find $\log_e \left( \frac{r}{s} \right)$

Now we have:

• $r = 2$
• $s = 2\sqrt{7}$

So, $\frac{r}{s} = \frac{2}{2\sqrt{7}} = \frac{1}{\sqrt{7}} = 7^{-\frac{1}{2}}$ Thus, $\log_e \left( \frac{r}{s} \right) = \log_e \left( 7^{-\frac{1}{2}} \right) = -\frac{1}{2} \log_e 7$

Final Answer

$\log_e \left( \frac{r}{s} \right) = -\frac{1}{2} \log_e 7$

Would you like more details or have any questions?

Here are some related questions:

1. How would the answer change if $\angle AOP$ were different?
2. What if $OP$ were perpendicular to $\angle AOP$ instead of $60^\circ$?
3. Can we generalize this approach for arbitrary values of $OP$, $\angle AOP$, $OC$?
4. What are the applications of intersection points in geometry problems?
5. How does the cosine rule help in determining side lengths in triangles?

Tip: For problems with intersecting circles, visualize key segments and use geometric properties like cosine rule for calculations.