[THOUGHT] We need to parse the problem carefully.

We have segment AB of length (10\sqrt{2}). Circle with diameter AB. Points P and Q lie on this circle. Given vector condition: [ (\overrightarrow{PA} + \overrightarrow{PB}) \cdot (\overrightarrow{PQ} + \overrightarrow{PB}) = 2 |\overrightarrow{PQ}|^2. ] Also (|\overrightarrow{PB}| = 14) and (|\overrightarrow{QB}| > 0). We need (|\overrightarrow{PA} \cdot \overrightarrow{QB}|) expressed as (\frac{q}{p}) with (p,q) coprime natural numbers. Then compute (p+q).

Let's set up coordinates.

Let’s place A and B conveniently. Let (A = (-5\sqrt{2}, 0)), (B = (5\sqrt{2}, 0)). Then AB length = (10\sqrt{2}). Circle with diameter AB: center at origin (0,0), radius (5\sqrt{2}).

Points P and Q lie on this circle. So coordinates: (P = (x_P, y_P)) with (x_P^2 + y_P^2 = (5\sqrt{2})^2 = 50). Similarly (Q = (x_Q, y_Q)) with (x_Q^2 + y_Q^2 = 50).

Given (|\overrightarrow{PB}| = 14). Vector PB = B - P = (5√2 - x_P, 0 - y_P). Its magnitude squared: ((5\sqrt{2} - x_P)^2 + y_P^2 = 14^2 = 196).

But P lies on circle: (x_P^2 + y_P^2 = 50).

Subtract: ((5\sqrt{2} - x_P)^2 + y_P^2 - (x_P^2 + y_P^2) = 196 - 50).

Expand: (50 - 10\sqrt{2}x_P + x_P^2 + y_P^2 - x_P^2 - y_P^2 = 146). Wait careful: ((5\sqrt{2} - x_P)^2 = (5\sqrt{2})^2 - 2\cdot 5\sqrt{2} x_P + x_P^2 = 50 - 10\sqrt{2} x_P + x_P^2).

Then ((5\sqrt{2} - x_P)^2 + y_P^2 = 50 - 10\sqrt{2} x_P + x_P^2 + y_P^2 = 50 - 10\sqrt{2} x_P + 50 = 100 - 10\sqrt{2} x_P).

Set equal to 196: (100 - 10\sqrt{2} x_P = 196 \Rightarrow -10\sqrt{2} x_P = 96 \Rightarrow x_P = -\frac{96}{10\sqrt{2}} = -\frac{48}{5\sqrt{2}} = -\frac{48\sqrt{2}}{10} = -\frac{24\sqrt{2}}{5}).

Thus (x_P = -\frac{24\sqrt{2}}{5}).

Then (y_P^2 = 50 - x_P^2 = 50 - \frac{1152}{25} = \frac{1250}{25} - \frac{1152}{25} = \frac{98}{25}). So (y_P = \pm \frac{7\sqrt{2}}{5}).

We'll keep both possibilities.

Now vector condition: ((\overrightarrow{PA} + \overrightarrow{PB}) \cdot (\overrightarrow{PQ} + \overrightarrow{PB}) = 2 |\overrightarrow{PQ}|^2).

Let’s express in terms of coordinates.

Let (P = (x_P, y_P)), (Q = (x_Q, y_Q)), (A = (-5\sqrt{2}, 0)), (B = (5\sqrt{2}, 0)).

Compute (\overrightarrow{PA} = A - P = (-5\sqrt{2} - x_P, -y_P)).

(\overrightarrow{PB} = B - P = (5\sqrt{2} - x_P, -y_P)).

Thus (\overrightarrow{PA} + \overrightarrow{PB} = (-5\sqrt{2} - x_P + 5\sqrt{2} - x_P, -y_P - y_P) = (-2x_P, -2y_P)).

So (\overrightarrow{PA} + \overrightarrow{PB} = -2P)? Actually -2 times P vector? Wait P is (x_P, y_P). So -2P = (-2x_P, -2y_P). Yes matches.

Now (\overrightarrow{PQ} = Q - P = (x_Q - x_P, y_Q - y_P)).

(\overrightarrow{PB} = (5\sqrt{2} - x_P, -y_P)).

Thus (\overrightarrow{PQ} + \overrightarrow{PB} = (x_Q - x_P + 5\sqrt{2} - x_P, y_Q - y_P - y_P) = (x_Q - 2x_P + 5\sqrt{2}, y_Q - 2y_P)).

Now dot product: ((\overrightarrow{PA} + \overrightarrow{PB}) \cdot (\overrightarrow{PQ} + \overrightarrow{PB}) = (-2x_P)(x_Q - 2x_P + 5\sqrt{2}) + (-2y_P)(y_Q - 2y_P)).

Right side: (2 |\overrightarrow{PQ}|^2 = 2[(x_Q - x_P)^2 + (y_Q - y_P)^2]).

Set them equal.

Also Q lies on circle: (x_Q^2 + y_Q^2 = 50).

We have unknowns (x_Q, y_Q) and known (x_P, y_P). Also (|\overrightarrow{QB}| > 0) meaning Q ≠ B, but that's fine.

We can attempt to solve for Q.

Let’s define variables in Python with sympy to solve.

But careful: The condition is a vector equation in 2D, giving one scalar equation. So there will be multiple solutions. But we also have the condition (|\overrightarrow{PA} \cdot \overrightarrow{QB}|) to compute, which might be independent of choice of Q? Possibly unique.

We can try solving for Q given P.

Let’s compute numerically.

We'll consider y_P positive (7√2/5) first, then negative maybe yields same magnitude for dot product? Might be symmetric.

Let’s proceed.

We'll use sympy to solve for Q coordinates satisfying both circle and the vector equation.

We'll also compute PB magnitude = 14 already used to find x_P.

Let’s code.

We'll set √2 as sqrt(2). Use sympy symbols.

We need to solve for Q = (x, y) with x^2 + y^2 = 50 and the dot product equation.

We'll also impose that Q is on circle.

Let’s do it.

We'll also compute PA · QB magnitude.

We'll see if we get a unique value for |PA · QB|.

We'll use rational approximations maybe, but sympy can handle exact.

Let’s code. [/PYTHON] [PYTHON] import sympy as sp

Define symbols

xP, yP, xQ, yQ = sp.symbols('xP yP xQ yQ', real=True)

Constants

sqrt2 = sp.sqrt(2) AB_half = 5sqrt2 radius_sq = (5sqrt2)**2 # 50

Given PB = 14

Equation: (5√2 - xP)^2 + yP^2 = 196

eq_PB = (5*sqrt2 - xP)2 + yP2 - 196

P lies on circle: xP^2 + yP^2 = 50

eq_circle_P = xP2 + yP2 - 50

Solve for xP, yP

sol_P = sp.solve([eq_PB, eq_circle_P], (xP, yP)) print("Solutions for P:", sol_P)

Should be two: xP = -24√2/5, yP = ±7√2/5

Let's compute exact values

xP_val = -24sqrt2/5 yP_val_pos = 7sqrt2/5 yP_val_neg = -7*sqrt2/5

We'll take yP positive first.

Define vector condition:

(PA + PB) · (PQ + PB) = 2 * |PQ|^2

PA = A - P, PB = B - P

A = sp.Point(-5sqrt2, 0) B = sp.Point(5sqrt2, 0) P = sp.Point(xP_val, yP_val_pos)

Define Q variables

x, y = sp.symbols('x y') Q = sp.Point(x, y)

Compute vectors

PA = sp.Matrix(A - P) PB = sp.Matrix(B - P) PQ = sp.Matrix(Q - P)

Left side: (PA + PB) · (PQ + PB)

left = (PA + PB).dot(PQ + PB)

Right side: 2 * |PQ|^2

right = 2 * PQ.dot(PQ)

Equation: left - right = 0

eq_vec = sp.simplify(left - right)

Q lies on circle: x^2 + y^2 = 50

eq_circle_Q = x2 + y2 - 50

Solve for x, y

solutions = sp.solve([eq_vec, eq_circle_Q], (x, y)) print("Solutions for Q (with yP positive):", solutions)

Might be multiple, possibly including P itself? But |QB| > 0 so Q ≠ B.

Let's filter Q ≠ B.

Now compute |PA · QB| for each solution

for sol in solutions: xq_val, yq_val = sol # Compute PA vector PA_vec = sp.Matrix(A - P) QB_vec = sp.Matrix(B - sp.Point(xq_val, yq_val)) dot_val = PA_vec.dot(QB_vec) abs_dot = sp.Abs(dot_val) print(f"Q = ({xq_val}, {yq_val})") print(f"PA · QB = {dot_val}") print(f"|PA · QB| = {abs_dot}") # Evaluate numerically print(f"Numerical |PA · QB| = {abs_dot.evalf()}")

Also try yP negative

P2 = sp.Point(xP_val, yP_val_neg) PA2 = sp.Matrix(A - P2) PB2 = sp.Matrix(B - P2) left2 = (PA2 + PB2).dot(PQ + PB2) right2 = 2 * PQ.dot(PQ) eq_vec2 = sp.simplify(left2 - right2) solutions2 = sp.solve([eq_vec2, eq_circle_Q], (x, y)) print("\nSolutions for Q (with yP negative):", solutions2) for sol in solutions2: xq_val, yq_val = sol PA2_vec = sp.Matrix(A - P2) QB2_vec = sp.Matrix(B - sp.Point(xq_val, yq_val)) dot_val2 = PA2_vec.dot(QB2_vec) abs_dot2 = sp.Abs(dot_val2) print(f"Q = ({xq_val}, {yq_val})") print(f"PA · QB = {dot_val2}") print(f"|PA · QB| = {abs_dot2}") print(f"Numerical |PA · QB| = {abs_dot2.evalf()}") [/PYTHON]

[THOUGHT] We need to find (|\overrightarrow{PA} \cdot \overrightarrow{QB}|) as (\frac{q}{p}) in lowest terms, then compute (p+q).

From the Python output, we have two possible values for (|\overrightarrow{PA} \cdot \overrightarrow{QB}|) depending on which Q is chosen (and possibly which P). But the problem says "Points P and Q lie on this circle and satisfy ..." and then gives (|\overrightarrow{PB}| = 14) and (|\overrightarrow{QB}| > 0). The condition (|\overrightarrow{QB}| > 0) excludes Q = B, but that doesn't exclude Q = (5√2, 0) which is point B? Wait, B is (5√2, 0). So Q = (5√2, 0) is exactly B, which would make (|\overrightarrow{QB}| = 0), violating (|\overrightarrow{QB}| > 0). So we must discard that solution.

Thus for yP positive, the only valid Q is ((527\sqrt{2}/125, -336\sqrt{2}/125)). Then (|\overrightarrow{PA} \cdot \overrightarrow{QB}| = 196/25). This is already in lowest terms: (p=25), (q=196), (p+q=221).

But wait: For yP negative, we have two solutions. The problem says "If (|\overrightarrow{PB}| = 14) and (|\overrightarrow{QB}| > 0)", meaning Q is not B. Both solutions in yP negative case give (|\overrightarrow{QB}| > 0) (since they are not B). However, the magnitude of (|\overrightarrow{PA} \cdot \overrightarrow{QB}|) is different for the two solutions: ~11.9146 and ~2.4095. Which one is correct? Possibly the problem expects a unique answer independent of choices. Maybe there is an additional implicit condition that picks one.

Let's examine the condition more carefully: ((\overrightarrow{PA} + \overrightarrow{PB}) \cdot (\overrightarrow{PQ} + \overrightarrow{PB}) = 2|\overrightarrow{PQ}|^2). This might force a specific configuration. Possibly the dot product magnitude is unique regardless of which Q (or P) we pick, but our numerical results show two different values for yP negative case. However, note that for yP negative, the two Q solutions are not symmetric; one yields ~11.9146, the other ~2.4095. So maybe we need to see which one is consistent with the geometry.

Alternatively, maybe the problem expects the answer from the yP positive case (since maybe P is uniquely determined by PB=14 and the circle, but yP could be ±). However, the problem doesn't specify orientation, so both P possibilities are allowed. But then the magnitude might be the same? Let's check: For yP positive, we got 196/25 = 7.84. For yP negative, we got two different numbers. So not unique unless we pick a specific Q.

Maybe we need to impose that Q is distinct from P and B, and maybe the dot product condition yields a unique Q up to symmetry. Possibly the problem expects the answer 196/25 because that's the only nice rational number.

Let's verify the calculation for yP positive case more carefully.

We had P = (-24√2/5, 7√2/5). Q = (527√2/125, -336√2/125). Compute PB magnitude: earlier we derived xP = -24√2/5, yP = ±7√2/5. Check PB magnitude: (5√2 - xP)^2 + yP^2 = (5√2 + 24√2/5)^2 + (7√2/5)^2 = ((25√2/5 + 24√2/5)^2) + (98/25) = ((49√2/5)^2) + 98/25 = (49^2 * 2 / 25) + 98/25 = (4802/25) + 98/25 = 4900/25 = 196. Yes correct.

Now compute PA · QB. Let's compute manually to confirm 196/25.

PA = A - P = (-5√2 - xP, -yP) = (-5√2 + 24√2/5, -7√2/5) = ((-25√2/5 + 24√2/5), -7√2/5) = (-√2/5, -7√2/5).

QB = B - Q = (5√2 - 527√2/125, 0 - (-336√2/125)) = ( (625√2/125 - 527√2/125), 336√2/125 ) = (98√2/125, 336√2/125).

Dot product: PA · QB = (-√2/5)(98√2/125) + (-7√2/5)(336√2/125) = (-298)/(5125) + (-72336)/(5*125) = (-196/625) + (-4704/625) = -4900/625 = -4900/625 = -4900÷25 / 625÷25 = -196/25. Yes matches.

Thus magnitude = 196/25.

Now, is this the only valid answer? The problem says "If |PB| = 14 and |QB| > 0, then |PA · QB| = q/p". It doesn't specify which Q, but maybe the condition forces Q to be that specific one. Possibly the vector condition plus Q on circle yields a unique Q (other than B) for each P. But we found two Q for yP positive: one is B (invalid), the other is the one we used. So for yP positive, only one valid Q.

For yP negative, there are two valid Q (both not B). But maybe the problem expects the answer independent of sign choice? That seems unlikely.

Maybe we need to consider that the vector condition might be symmetric under reflection across x-axis? If we take P with yP negative, the Q solutions might be reflections of the Q solutions for yP positive? Let's check: For yP positive, Q = (527√2/125, -336√2/125). For yP negative, one Q is ( -49√15658/1695 + 4016√2/1695, -56√15658/1695 - 3514√2/1695 ) which is not simply the reflection. So not symmetric.

Maybe the problem expects the answer 196/25. That gives p+q = 25+196 = 221.

Let's check if 196/25 is in lowest terms: gcd(196,25)=1. Yes.

Thus p+q = 221.

But wait: Could there be another valid Q for yP positive? We discarded Q = (5√2, 0) because |QB| = 0. So only one valid Q.

Thus answer likely 221.

Let's double-check the condition |QB| > 0: For Q = (5√2, 0), QB = B - Q = (0,0), magnitude 0, so violates. So discard.

Thus answer is 196/25.

Thus p+q = 221.

We'll answer 221.

But before finalizing, let's verify that the vector condition indeed yields only those Q. Possibly we made an algebra mistake? Let's quickly re-derive the condition symbolically to ensure no error.

We can re-derive the vector condition in terms of coordinates.

We had: (PA+PB) = -2P. (PQ+PB) = (xQ - 2xP + 5√2, yQ - 2yP).

Dot product: -2xP*(xQ - 2xP + 5√2) - 2yP*(yQ - 2yP) = 2[(xQ - xP)^2 + (yQ - yP)^2].

Divide by 2: -xP*(xQ - 2xP + 5√2) - yP*(yQ - 2yP) = (xQ - xP)^2 + (yQ - yP)^2.

Expand left: -xP xQ + 2xP^2 - 5√2 xP - yP yQ + 2yP^2.

Right: xQ^2 - 2xP xQ + xP^2 + yQ^2 - 2yP yQ + yP^2.

Bring all to one side:

Left - right = (-xP xQ + 2xP^2 - 5√2 xP - yP yQ + 2yP^2) - (xQ^2 - 2xP xQ + xP^2 + yQ^2 - 2yP yQ + yP^2) = 0.

Simplify: -xP xQ + 2xP^2 - 5√2 xP - yP yQ + 2yP^2 - xQ^2 + 2xP xQ - xP^2 - yQ^2 + 2yP yQ - yP^2 = 0.

Combine xP xQ terms: (-xP xQ + 2xP xQ) = xP xQ.

Combine xP^2 terms: 2xP^2 - xP^2 = xP^2.

Similarly yP yQ: -yP yQ + 2yP yQ = yP yQ.

yP^2: 2yP^2 - yP^2 = yP^2.

Thus: xP xQ - 5√2 xP + yP yQ + xP^2 + yP^2 - xQ^2 - yQ^2 = 0.

But xP^2 + yP^2 = 50. So:

xP xQ - 5√2 xP + yP yQ + 50 - (xQ^2 + yQ^2) = 0.

Since xQ^2 + yQ^2 = 50, we get:

xP xQ - 5√2 xP + yP yQ + 50 - 50 = 0 ⇒ xP xQ - 5√2 xP + yP yQ = 0.

Thus condition simplifies to:

xP xQ + yP yQ = 5√2 xP.

Interesting! That's much simpler.

So vector condition reduces to dot product P · Q = 5√2 xP.

Given P known, this is a linear equation in Q.

Also Q lies on circle xQ^2 + yQ^2 = 50.

So intersection of line (dot product) with circle. That yields up to two solutions.

Now with P = (-24√2/5, 7√2/5), xP = -24√2/5.

Then 5√2 xP = 5√2 * (-24√2/5) = -24 * 2 = -48.

So condition: xP xQ + yP yQ = -48.

Compute: (-24√2/5) xQ + (7√2/5) yQ = -48.

Multiply by 5: -24√2 xQ + 7√2 yQ = -240.

Divide by √2: -24 xQ + 7 yQ = -240/√2 = -120√2.

Thus line: -24 xQ + 7 yQ = -120√2.

And circle: xQ^2 + yQ^2 = 50.

Solve. Let's solve quickly to verify Q.

We can do in Python quickly.

But we already have solutions: (5√2, 0) and (527√2/125, -336√2/125). Let's check if they satisfy line.

For Q = (5√2, 0): -24*(5√2) + 7*0 = -120√2, matches.

For Q = (527√2/125, -336√2/125): -24*(527√2/125) + 7*(-336√2/125) = (-12648√2/125) + (-2352√2/125) = -15000√2/125 = -120√2. Yes.

So both are valid.

Now for yP negative: P = (-24√2/5, -7√2/5). Then 5√2 xP = -48 same. Condition: xP xQ + yP yQ = -48.

Compute: (-24√2/5) xQ + (-7√2/5) yQ = -48 ⇒ multiply by 5: -24√2 xQ -7√2 yQ = -240 ⇒ divide √2: -24 xQ -7 yQ = -120√2.

So line: -24 xQ -7 yQ = -120√2.

Intersect with circle yields two solutions. That matches the two solutions we got earlier (messy radicals).

Now, the problem says |QB| > 0, so Q ≠ B. For yP positive, B is one solution, so the other is the only valid Q. For yP negative, B is not a solution? Let's check if B satisfies line: B = (5√2, 0). Plug into -24*(5√2) -70 = -120√2, yes it satisfies. So B is also a solution for yP negative? Wait line is -24 xQ -7 yQ = -120√2. For B: -24(5√2) = -120√2, yes. So B is also a solution for yP negative. But B is not on the circle? Wait B is (5√2, 0) and 5√2 squared = 50, yes B is on circle. So B is also a solution for yP negative. However, earlier Python output for yP negative gave two solutions that are not B. That suggests maybe we missed B? Let's check: The line equation for yP negative: -24 x -7 y = -120√2. Circle: x^2 + y^2 = 50. Solve: one solution is B = (5√2, 0). The other solution should be something else. But our Python gave two solutions that are not B. Possibly because Python solved the system and gave two solutions, one of which might be B? Let's check if B satisfies the second solution? The first solution: x = -49√15658/1695 + 4016√2/1695, y = -56√15658/1695 - 3514√2/1695. That's not B. The second solution also not B. So B is not among them. That means maybe our line equation derivation is wrong? Let's re-derive carefully.

We had condition: xP xQ + yP yQ = 5√2 xP.

For P = (xP, yP) = (-24√2/5, -7√2/5). Then 5√2 xP = 5√2 * (-24√2/5) = -24*2 = -48.

Left: xP xQ + yP yQ = (-24√2/5) xQ + (-7√2/5) yQ = (-24√2 xQ -7√2 yQ)/5.

Set equal to -48: (-24√2 xQ -7√2 yQ)/5 = -48 ⇒ multiply by 5: -24√2 xQ -7√2 yQ = -240 ⇒ divide by √2: -24 xQ -7 yQ = -240/√2 = -120√2. Yes.

Now B = (5√2, 0): LHS = -24*(5√2) -7*0 = -120√2, matches. So B should be a solution. Why didn't Python find it? Possibly because when solving, sympy might have simplified and missed B? Let's check by solving directly with the line and circle.

We'll do a quick check in Python. [/PYTHON] [PYTHON] import sympy as sp

P with y negative

xP, yP = -24sp.sqrt(2)/5, -7sp.sqrt(2)/5

Line: xPx + yPy = 5*sqrt(2)*xP

But 5*sqrt(2)*xP = -48

So equation: xPx + yPy = -48

line_eq = sp.Eq(xPx + yPy, -48)

Circle

circle_eq = sp.Eq(x2 + y2, 50)

Solve

solutions = sp.solve([line_eq, circle_eq], (x, y)) print("Solutions for Q (yP negative):", solutions)

Check if (5*sqrt(2), 0) is among them

B = sp.Point(5sp.sqrt(2), 0) print("Is B a solution?", (solutions[0][0].subs(x,5sp.sqrt(2)).subs(y,0) == 0 and solutions[0][1].subs(x,5sp.sqrt(2)).subs(y,0) == 0) or (solutions[1][0].subs(x,5sp.sqrt(2)).subs(y,0) == 0 and solutions[1][1].subs(x,5*sp.sqrt(2)).subs(y,0) == 0))

Evaluate each solution

for sol in solutions: print("Solution:", sol) print("Check line:", (xPsol[0] + yPsol[1]).evalf()) print("Check circle:", (sol[0]**2 + sol[1]**2).evalf()) [/PYTHON]

[THOUGHT] Now we have the correct solutions: For yP negative, the two Q solutions are (527√2/125, 336√2/125) and (5√2, 0). The latter is B, which is invalid because |QB| = 0. So only valid Q is (527√2/125, 336√2/125).

Wait, earlier for yP positive we got Q = (527√2/125, -336√2/125). For yP negative, we get Q = (527√2/125, 336√2/125). So Q's y-coordinate sign flips with P's y-coordinate sign.

Now compute |PA · QB| for this Q.

Let's compute quickly with Python to confirm. [/PYTHON] [PYTHON] import sympy as sp

sqrt2 = sp.sqrt(2)

P with y negative

xP = -24sqrt2/5 yP = -7sqrt2/5 P = sp.Point(xP, yP) A = sp.Point(-5sqrt2, 0) B = sp.Point(5sqrt2, 0) Q = sp.Point(527sqrt2/125, 336sqrt2/125)

PA = sp.Matrix(A - P) QB = sp.Matrix(B - Q) dot_val = PA.dot(QB) abs_dot = sp.Abs(dot_val) print("PA · QB =", dot_val) print("|PA · QB| =", abs_dot) print("Numerical:", abs_dot.evalf())

Simplify

abs_dot_simplified = sp.simplify(abs_dot) print("Simplified:", abs_dot_simplified) print("As fraction:", sp.nsimplify(abs_dot_simplified)) [/PYTHON]

[THOUGHT] Thus, regardless of whether P is above or below the x-axis, the valid Q (excluding B) yields |PA · QB| = 196/25. Therefore p = 25, q = 196, and p + q = 221.

Final answer: \boxed{221}.