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 .
[THOUGHT] The locus length k is given by the integral along R of 9 / |AP|^2 ds. R consists of two arcs.
For the C1 arc (θ from 2π/3 to 4π/3): |AP|^2 = 5 + 4 cos θ, ds = dθ. Split at θ=π: integral of 1/(5+4 cos θ) is 4π/9, times 9 is 4π.
For the C2 arc (φ from -π/3 to π/3): |AP|^2 = 2(1 + cos φ), ds = dφ. Simplifies to (9/4) sec^2(φ/2) dφ, integrates to 3√3.
Thus k = 4π + 3√3. To compute floor(k), evaluate numerically. [/THOUGHT] [PYTHON] import math k = 4 * math.pi + 3 * math.sqrt(3) print(k) print(math.floor(k)) [/PYTHON]
[THOUGHT] The Python output gives k ≈ 17.762523037065804, so ⌊k⌋ = 17. [/THOUGHT] \boxed{17}
Explanation
The reasoning derives the locus length k by integrating the expression 9 / |AP|^2 along the region R, which comprises two circular arcs.
For the arc of C1 from θ = 2π/3 to 4π/3 (passing through O'(-1,0)), |AP|^2 = 5 + 4 cos θ and ds = dθ. The integral ∫ 9 / (5 + 4 cos θ) dθ over this interval evaluates to 4π, using the standard integral form for 1 / (a + b cos θ).
For the arc of C2 from φ = -π/3 to π/3 (passing through O(0,0)), |AP|^2 = 2(1 + cos φ) and ds = dφ. Substituting yields ∫ (9/4) sec^2(φ/2) dφ from -π/3 to π/3, which integrates to 3√3.
Thus, k = 4π + 3√3. Numerical evaluation via Python confirms k ≈ 17.762523037065804, so ⌊k⌋ = 17, as directly output by math.floor(k).