Problem Analysis #49

Based on the problem statement, I will perform the following steps using Python:

1. Define the radius $R = 5\sqrt{2}$.
2. Define point $B = (R, 0)$ and point $A = (-R, 0)$.
3. Determine the coordinates of point $P$. We know $|P - B| = 14$ and $|P| = R$.
   • Let $P = (x_p, y_p)$.
   • Solve $(x_p - R)^2 + y_p^2 = 14^2$ and $x_p^2 + y_p^2 = R^2$.
4. Determine the coordinates of point $Q$.
   • The condition $(\overrightarrow{PA} + \overrightarrow{PB}) \cdot (\overrightarrow{PQ} + \overrightarrow{PB}) = 2|\overrightarrow{PQ}|^2$ simplifies to $\overrightarrow{OP} \cdot \overrightarrow{BQ} = 0$, which implies $P \cdot Q = P \cdot B$.
   • Solve the system $x_p x_q + y_p y_q = x_p R$ (since $y_B=0$) and $x_q^2 + y_q^2 = R^2$.
   • This will yield two solutions for $Q$. One should be $B$ (where $|QB|=0$). We select the other one.
5. Calculate the value $|\overrightarrow{PA} \cdot \overrightarrow{QB}|$.
   • $\overrightarrow{PA} = A - P$.
   • $\overrightarrow{QB} = B - Q$.
   • Compute the dot product and take the absolute value.
6. Express the result as a fraction $q/p$ and find $p+q$.

[PYTHON] import math from sympy import symbols, solve, Rational, sqrt, simplify

def solve_problem(): # Define R # R^2 = 50 R_sq = 50 R = math.sqrt(R_sq)

# Coordinates of B
xB, yB = R, 0

# Find P (xp, yp)
# (xp - xB)^2 + yp^2 = 14^2 = 196
# xp^2 + yp^2 = 50
# xp^2 - 2*xp*xB + xB^2 + yp^2 = 196
# 50 - 2*xp*R + 50 = 196
# 100 - 2*xp*R = 196
# -2*xp*R = 96
# xp = -48 / R

xp = -48 / R

# Check if valid
if xp**2 > R_sq:
    print("Error: P not on circle")
    return
    
yp_abs = math.sqrt(R_sq - xp**2)
# We can choose positive yp, by symmetry the result should be the same
yp = yp_abs

# Verify PB length
dist_PB = math.sqrt((xp - xB)**2 + (yp - yB)**2)
print(f"PB length check: {dist_PB} (expected 14)")

# Find Q (xq, yq)
# Condition derived: P . Q = P . B
# xp*xq + yp*yq = xp*xB + yp*yB
# xp*xq + yp*yq = xp*R

dot_PB = xp * xB + yp * yB

# We need to solve for xq, yq:
# 1) xp*xq + yp*yq = dot_PB
# 2) xq^2 + yq^2 = R_sq

# Using sympy for exact arithmetic to get fraction
# Let's switch to exact values

# R = 5*sqrt(2)
# xp = -48 / (5*sqrt(2)) = -48*sqrt(2)/10 = -24*sqrt(2)/5

s_R = 5 * sqrt(2)
s_xB = s_R
s_yB = 0
s_xp = -48 / s_R
s_yp = sqrt(50 - s_xp**2) # This might be complex if we aren't careful, but we checked values

# s_xp = -24*sqrt(2)/5
# s_xp^2 = 576*2/25 = 1152/25 = 46.08
# 50 - 46.08 = 3.92 = 392/100 = 98/25
# sqrt(98/25) = 7*sqrt(2)/5

s_xp = -Rational(24, 5) * sqrt(2)
s_yp = Rational(7, 5) * sqrt(2)

s_dot_PB = s_xp * s_xB + s_yp * s_yB # = -24/5 * sqrt(2) * 5 * sqrt(2) = -24 * 2 = -48

# Solve for Q
xq, yq = symbols('xq yq', real=True)
eq1 = s_xp * xq + s_yp * yq - s_dot_PB
eq2 = xq**2 + yq**2 - 50

solutions = solve([eq1, eq2], (xq, yq))

print(f"Solutions for Q: {solutions}")

# Identify the correct Q
# One solution should be B (5*sqrt(2), 0)
# We need the other one

valid_Q = None
for sol in solutions:
    sxq, syq = sol
    # Check distance to B
    dist_sq = (sxq - s_xB)**2 + (syq - s_yB)**2
    if simplify(dist_sq) > 0:
        valid_Q = (sxq, syq)
        break
        
if valid_Q is None:
    print("No valid Q found (distinct from B)")
    return

s_xq, s_yq = valid_Q
print(f"Selected Q: {valid_Q}")

# Calculate target: |PA . QB|
# A = (-R, 0)
s_xA = -s_R
s_yA = 0

# Vector PA = A - P
v_PA_x = s_xA - s_xp
v_PA_y = s_yA - s_yp

# Vector QB = B - Q
v_QB_x = s_xB - s_xq
v_QB_y = s_yB - s_yq

dot_prod = v_PA_x * v_QB_x + v_PA_y * v_QB_y
result = abs(dot_prod)

print(f"Result (unsimplified): {result}")
print(f"Result (simplified): {simplify(result)}")

solve_problem() [/PYTHON]

The problem asks for $p+q$ where $\left|\overrightarrow{\text{PA}} \cdot \overrightarrow{\text{QB}}\right| = \dfrac{q}{p}$. Based on the Python code execution:

1. The radius of the circle is $R = 5\sqrt{2}$, so $R^2 = 50$.
2. Point $B$ is at $(5\sqrt{2}, 0)$ and $A$ is at $(-5\sqrt{2}, 0)$.
3. Point $P$ satisfies $|PB|=14$, leading to $P \cdot B = -48$.
4. The vector condition implies $P \cdot Q = P \cdot B = -48$.
5. Solving for $Q$ on the circle (distinct from $B$) gives the coordinates used in the code.
6. The dot product $\overrightarrow{PA} \cdot \overrightarrow{QB}$ was calculated as $-\frac{196}{25}$.
7. Taking the absolute value gives $\frac{196}{25}$.

Thus, the fraction is $\frac{196}{25}$. $q = 196$ and $p = 25$. These numbers are coprime ($196 = 2^2 \cdot 7^2$, $25 = 5^2$). The sum $p + q = 25 + 196 = 221$.

$\boxed{221}$