Consider the point in the coordinate plane. Suppose two points and satisfy the following conditions:
- [(i)] The vectors and are parallel.
- [(ii)] The dot product satisfies . Let the two circlesintersect at two distinct points and .Consider the region consisting of the arc of circle ,which contains the point , together with the arc of circle ,which contains the point .When the point moves along the region , let denote the total lengthof the locus traced by the corresponding point .For any real number , let denote the greatest integer less than or equal to .Determine the value of .
The locus of point (Q) is traced as (P) moves along the region (\mathcal{R}), which consists of two arcs: the arc (R_1O'R_2) on circle (C_1) and the arc (R_1OR_2) on circle (C_2).
For (P) on the arc (R_1O'R_2) of (C_1) (parameterized by (\theta) from (120^\circ) to (240^\circ)), the locus of (Q) is an arc of the circle ((x-4)^2 + y^2 = 9). The arc length is (6\pi\sqrt{3}).
For (P) on the arc (R_1OR_2) of (C_2) (parameterized by (\phi) from (60^\circ) to (-60^\circ)), the locus of (Q) is a vertical line segment at (x = 2.5) from ((2.5, \frac{3\sqrt{3}}{2})) to ((2.5, -\frac{3\sqrt{3}}{2})). The length of this segment is (3\sqrt{3}).
The total length (k) of the locus is the sum of these two lengths: [ k = 6\pi\sqrt{3} + 3\sqrt{3} = 3\sqrt{3}(2\pi + 1). ]
Numerically, (2\pi \approx 6.283185307), so (2\pi + 1 \approx 7.283185307). Then (\sqrt{3} \approx 1.73205080757), so (3\sqrt{3} \approx 5.19615242271). Thus, [ k \approx 5.19615242271 \times 7.283185307 \approx 37.84454. ]
The floor of (k) is (\lfloor k \rfloor = 37).
\boxed{37}
[THOUGHT] We need to compute the total length k using Python to get the numerical value and then take the floor. [/THOUGHT] [PYTHON] import math k = 3 * math.sqrt(3) * (2 * math.pi + 1) k_floor = math.floor(k) print(k_floor) [/PYTHON]
\boxed{37}