Problem Analysis #29

[THOUGHT] To solve this problem, we need to understand the movement of point (Q) relative to point (P) as (P) moves along the region (\mathcal{R}).

Step 1: Condition Analysis

• (\overrightarrow{AP}) and (\overrightarrow{AQ}) are parallel, so (\overrightarrow{AQ} = \lambda \overrightarrow{AP}) for some scalar (\lambda).
• Given (\overrightarrow{AP} \cdot \overrightarrow{AQ} = 9), we have (\overrightarrow{AP} \cdot (\lambda \overrightarrow{AP}) = \lambda |\overrightarrow{AP}|^2 = 9). Therefore, (\lambda = \frac{9}{|\overrightarrow{AP}|^2}).

Step 2: Region (\mathcal{R}) Analysis

• (\mathcal{R}) consists of two arcs:
  • (R_1O'R_2) on (C_1) (circle centered at origin with radius 1)
  • (R_1OR_2) on (C_2) (circle centered at ((-1,0)) with radius 1)
• These circles intersect at two points, which we need to find.

Step 3: Finding Intersection Points

• (C_1): (x^2 + y^2 = 1)
• (C_2): ((x+1)^2 + y^2 = 1)
• Expanding (C_2) gives (x^2 + 2x + 1 + y^2 = 1), which simplifies to (x^2 + y^2 + 2x = 0).
• Using (x^2 + y^2 = 1) from (C_1), we substitute to get (1 + 2x = 0) or (x = -\frac{1}{2}).
• Substituting (x = -\frac{1}{2}) into (C_1), we get (\left(-\frac{1}{2}\right)^2 + y^2 = 1), or (\frac{1}{4} + y^2 = 1), hence (y^2 = \frac{3}{4}), so (y = \pm \frac{\sqrt{3}}{2}).
• Thus, the intersection points are (R_1 = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)) and (R_2 = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)).

Step 4: Calculate Length of Arcs

• The arc (R_1O'R_2) on (C_1) is a 120° arc (since the angle (R_1OR_2) is 120°, given symmetry and equal radii). The length is (\frac{120}{360} \times 2\pi \times 1 = \frac{1}{3} \times 2\pi = \frac{2\pi}{3}).
• Similarly, the arc (R_1OR_2) on (C_2) is also a 120° arc. The length is (\frac{120}{360} \times 2\pi \times 1 = \frac{2\pi}{3}).
• Total arc length for (P) along (\mathcal{R}) is (\frac{2\pi}{3} + \frac{2\pi}{3} = \frac{4\pi}{3}).

Step 5: Locus of (Q)

• As (P) moves on an arc, (\overrightarrow{AQ} = \frac{9}{|\overrightarrow{AP}|^2} \overrightarrow{AP}).
• Since (P) moves on a circular arc around (A), (Q) will also move on a circular arc around (A), but with a varying radius determined by (|\overrightarrow{AP}|). However, the total angle covered by (Q) will be the same as (P) because the direction of (\overrightarrow{AP}) and (\overrightarrow{AQ}) is the same.
• Thus, (Q) traces the same 120° arc twice on two different circles, each proportional to the distance from (A).

Step 6: Calculate the Length of Locus (k) for (Q)

• The length of each arc traced by (Q) is proportional to the length of the arc traced by (P). Since (Q) is effectively moving at the same angular speed as (P), the total length (k) of the locus traced by (Q) is the same as the total length of the locus traced by (P), which is (\frac{4\pi}{3}).

Step 7: Final Calculation

• The value of (k) is (\frac{4\pi}{3} \approx 4.1888).
• (\lfloor k \rfloor = \lfloor 4.1888 \rfloor = 4).

[/THOUGHT] \boxed{4}