Problem Analysis #29

To determine the value of $\lfloor k \rfloor$, we analyze the problem by considering the transformation from point $P$ to point $Q$ under the given geometric constraints.

Step 1: Understanding the Transformation

We are given:

• Point $A = (-2, 0)$
• Points $P$ and $Q$ satisfy:
  • $\overrightarrow{AP} \parallel \overrightarrow{AQ}$
  • $\overrightarrow{AP} \cdot \overrightarrow{AQ} \cdot AQ = 9$

This implies that vector $\overrightarrow{AQ}$ is a scalar multiple of $\overrightarrow{AP}$, and since their dot product is positive, they point in the same direction. Thus,

So the transformation from $P = (x, y)$ to $Q = (X, Y)$ is:

This is a circle inversion centered at $A = (-2, 0)$ with radius squared 9.

Step 2: Region $R$ and Its Image Under Inversion

The region $R$ is defined as the union of two circular arcs:

• Arc 1: On circle $C_1: x^2 + y^2 = 1$, from $R_1 = (-1/2, \sqrt{3}/2)$ to $R_2 = (-1/2, -\sqrt{3}/2)$, passing through $O' = (-1, 0)$.
• Arc 2: On circle $C_2: (x+1)^2 + y^2 = 1$, from $R_1$ to $R_2$, passing through $O = (0, 0)$.

These arcs form a closed curve, forming the boundary of region $R$.

We now compute the length of the image of this boundary under the inversion transformation.

Step 3: Length of Image of Arc 1

Arc 1 is part of circle $C_1: x^ with parameters$\theta \in [2\pi/3, 4\pi/3]$, and the image of$P$under the inversion is a circular arc on the inverse circle of$C_1$, which has:

• Center at $(4, 0)$
• Radius $r = 3$

The arc corresponds to a central angle of $2\pi/3$ (120°), so the arc length is:

---a.e.

Step 4: Length of Image of Arc 2

Arc 2 lies on circle $C_2: (x+1)^2 + y^2 = 1$, which passes through point $A = (-2, 0)$. Therefore, the image of this arc under the inversion is a line segment.

Through direct computation (or using known properties of inversion), we find that the image of Arc 2 lies on the vertical line:

with $Y \in \left[ -\frac{3\sqrt{3}}{2}, \frac{3\sqrt{3}}{2} \right]$

Thus, the length of the image of Arc 2 is:

Step 5: Total Length $k$

Approximating:

• $2\pi \approx 6.283$
• $3\sqrt{3} \ with right parenthesis 5.196$

Final Step: Compute $\lfloor k \rfloor$

[THOUGHT] To determine the value of $\lfloor k \rfloor$, we analyze the geometric transformation from the region $R$ to the point $Q$ using circle inversion. The key steps involve calculating the lengths of the image arcs under this inversion.

Step 1: Understand the Inversion

• The transformation is defined such that $\overrightarrow{AQ} = \frac{9}{|\overrightarrow{AP}|^2} \overrightarrow{AP}$, which is a circle inversion centered at $A(-2, 0)$ with radius squared 9.
• Region $R$ is bounded by two circular arcs:
  • Arc 1: On circle $C_1: x^2 + y^2 = 1$, from $R_1 = (-1/2, \sqrt{3}/2)$ to $R_2 = (-1/2, -\sqrt{3}/2)$, passing through $O' = (-1, 0)$.
  • Arc 2: On circle $C_2: (x+1)^2 + y^2 = 1$, from $R_1$ to $R_2$, passing through $O = (0, 0)$.

Step 2: Image of Arc 1 (on $C_1$)

• The image of $C_1$ under the inversion is a circle with center $(4, 0)$ and radius 3.
• The arc on $C_1$ corresponds to a central angle of $2\pi/3$ radians in the image.
• Length of image arc: $3 \times (2\pi/3) = 2\pi$.

Step 3: Image of Arc 2 (on $C_2$)

• The image of $C_2$ (which passes through $A$) is a vertical line $X = 5/2$.
• The arc on $C_2$ corresponds to a line segment from $(5/2, 3\sqrt{3}/2)$ to $(5/2, -3\sqrt{3}/2)$.
• Length of image segment: $3\sqrt{3}$.

Step 4: Total Length $k$

Verification via Python: [PYTHON] import math

k = 2 * math.pi + 3 * math.sqrt(3) print(k) print(math.floor(k)) [/PYTHON]

The output is:

11.479190432858417
11

Final Answer

[THOUGHT] \boxed{11}