[THOUGHT] We have a circle with diameter AB, so AB is a diameter, length = 10√2. So radius = (10√2)/2 = 5√2. Center O is midpoint of AB. Points P and Q lie on the circle. Given |PB| = 14. Also given vector condition: (PA + PB)·(PQ + PB) = 2|PQ|^2. We need |PA·QB| = q/p, find p+q.

Let's set up coordinates conveniently. Let A = (-r,0), B = (r,0) where r = 5√2. Then AB length = 2r = 10√2 indeed. Center O at (0,0). Circle equation: x^2 + y^2 = r^2 = (5√2)^2 = 50.

Given |PB| = 14. Let P = (x_p, y_p) on circle: x_p^2 + y_p^2 = 50. Distance PB = sqrt((x_p - r)^2 + y_p^2) = 14. We can solve for P coordinates.

Also Q is on circle: x_q^2 + y_q^2 = 50.

Vector condition: (PA + PB)·(PQ + PB) = 2|PQ|^2.

We can simplify vectors: PA = A - P, PB = B - P, PQ = Q - P.

Compute PA + PB = (A - P) + (B - P) = (A + B) - 2P = (0,0) - 2P = -2P because A+B = (0,0) since symmetric about origin. Indeed A = (-r,0), B = (r,0), sum = (0,0). So PA + PB = -2P.

PQ + PB = (Q - P) + (B - P) = Q + B - 2P.

Thus LHS = (-2P)·(Q + B - 2P) = -2P·Q - 2P·B + 4P·P.

But P·P = |P|^2 = r^2 = 50.

So LHS = -2P·Q - 2P·B + 200.

RHS = 2|PQ|^2 = 2|Q - P|^2 = 2[(Q-P)·(Q-P)] = 2(Q·Q - 2Q·P + P·P) = 2(r^2 - 2Q·P + r^2) = 2(100 - 2Q·P) = 200 - 4Q·P.

Set LHS = RHS: -2P·Q - 2P·B + 200 = 200 - 4P·Q => -2P·Q - 2P·B = -4P·Q => Add 4P·Q both sides: -2P·Q - 2P·B + 4P·Q = 0 => 2P·Q - 2P·B = 0 => P·Q - P·B = 0 => P·Q = P·B.

Thus condition simplifies to P·Q = P·B.

Now we know |PB| = 14, and P·B = dot product P with B = (r,0). P = (x_p, y_p), B = (r,0) => P·B = r x_p.

Also P·Q = x_p x_q + y_p y_q.

So condition: x_p x_q + y_p y_q = r x_p. (1)

Also Q on circle: x_q^2 + y_q^2 = 50. (2)

We also have P on circle: x_p^2 + y_p^2 = 50. (3)

And distance PB = sqrt((x_p - r)^2 + y_p^2) = 14. Square: (x_p - r)^2 + y_p^2 = 196. Expand: x_p^2 - 2r x_p + r^2 + y_p^2 = 196. But x_p^2 + y_p^2 = 50, r^2 = 50. So 50 - 2r x_p + 50 = 196 => 100 - 2r x_p = 196 => -2r x_p = 96 => r x_p = -48 => x_p = -48/r.

Compute r = 5√2 ≈ 7.071..., but keep exact: r = 5√2.

x_p = -48/(5√2) = -48√2/(10) = -24√2/5.

Then y_p^2 = 50 - x_p^2 = 50 - (24√2/5)^2 = 50 - (5762)/25 = 50 - 1152/25 = (1250 - 1152)/25 = 98/25. So y_p = ±√(98/25) = ±(7√2)/5? Let's check: √98 = √(492)=7√2, over 5 indeed. So y_p = ±(7√2)/5.

We need sign? Possibly both possible, but we'll keep variable sign.

Now P·B = r x_p = -48 (we computed earlier). So P·Q = -48.

Thus x_p x_q + y_p y_q = -48. (1')

We want |PA·QB|. PA = A - P = (-r - x_p, 0 - y_p) = (-r - x_p, -y_p). QB = B - Q = (r - x_q, 0 - y_q). Dot product PA·QB = (-r - x_p)(r - x_q) + (-y_p)(-y_q) = (-r - x_p)(r - x_q) + y_p y_q.

We need absolute value of that dot product.

We can compute numerically with Python, keeping exact symbolic maybe using sympy.

Let's set up sympy to solve for Q coordinates from (1') and (2), then compute PA·QB, take absolute value, simplify to fraction q/p.

We'll need to consider both signs for y_p. But maybe sign doesn't affect absolute value? Possibly symmetry. Let's compute both.

We'll also note that |QB| > 0, so Q ≠ B.

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

define symbols

r = 5*sp.sqrt(2)

P coordinates

x_p = -48/r # simplify x_p_simplified = sp.simplify(x_p) print("x_p =", x_p_simplified)

compute y_p^2

y_p_sq = 50 - x_p_simplified**2 print("y_p^2 =", sp.simplify(y_p_sq)) y_p = sp.sqrt(y_p_sq) # positive root, but we can consider ± print("y_p =", sp.simplify(y_p))

Actually y_p = ± sqrt(98)/5? Let's compute exact.

Let's compute numeric approximations for clarity.

print("x_p numeric:", x_p_simplified.evalf()) print("y_p numeric:", y_p.evalf())

We'll keep symbolic with sqrt(2).

Let's define y_p as ± (7*sqrt(2))/5.

y_p_pos = 7sp.sqrt(2)/5 y_p_neg = -7sp.sqrt(2)/5 print("y_p_pos =", y_p_pos) print("y_p_neg =", y_p_neg)

Verify that x_p^2 + y_p^2 = 50

print("Check circle for pos:", sp.simplify(x_p_simplified2 + y_p_pos2)) print("Check circle for neg:", sp.simplify(x_p_simplified2 + y_p_neg2))

Good.

Now Q coordinates variables

x_q, y_q = sp.symbols('x_q y_q', real=True)

Equations: (1) x_px_q + y_py_q = -48, (2) x_q^2 + y_q^2 = 50

Solve for x_q, y_q for each y_p sign.

for y_p_val, label in [(y_p_pos, 'y_p positive'), (y_p_neg, 'y_p negative')]: print("\n---", label, "---") eq1 = x_p_simplifiedx_q + y_p_valy_q + 48 # =0 eq2 = x_q2 + y_q2 - 50 sols = sp.solve([eq1, eq2], (x_q, y_q), dict=True) print("Solutions for Q:", sols) # For each solution, compute PA·QB and absolute value for sol in sols: x_q_val = sol[x_q] y_q_val = sol[y_q] # PA = A - P = (-r - x_p, -y_p) PA_x = -r - x_p_simplified PA_y = -y_p_val # QB = B - Q = (r - x_q, -y_q) QB_x = r - x_q_val QB_y = -y_q_val dot = PA_xQB_x + PA_yQB_y dot_simplified = sp.simplify(dot) abs_dot = sp.Abs(dot_simplified) print("Q =", (x_q_val, y_q_val)) print("PA·QB =", dot_simplified) print("|PA·QB| =", abs_dot) # Evaluate numeric to see fraction print("Numeric value:", abs_dot.evalf()) # Try to express as rational sqrt? Maybe it's rational. # Let's see if dot_simplified is rational. print("Is dot rational?", dot_simplified.is_rational) # If not, maybe square? print("Square of dot:", sp.simplify(dot_simplified2)) # We need q/p where p,q coprime naturals => value is rational. # So |PA·QB| must be rational. Let's check. # Compute exact rational approximation? # Use sp.nsimplify with rationals. rat = sp.nsimplify(abs_dot, [sp.sqrt(2)]) print("Rational approximation:", rat) # If rat is Rational, get numerator/denominator. if rat.is_Rational: p = rat.q q = rat.p print("p =", p, "q =", q) print("p+q =", p+q) else: # maybe it's a rational multiple of sqrt(2)? But answer is q/p rational. # Possibly absolute value eliminates sqrt. # Let's compute numeric and see if it's rational. num_val = abs_dot.evalf(30) print("High precision:", num_val) # Check if close to rational. from fractions import Fraction # approximate as fraction frac = Fraction(num_val).limit_denominator(10000) print("Fraction approx:", frac) # But we need exact. # Let's compute symbolically. # Compute dot^2 and see if it's rational. dot_sq = sp.simplify(dot_simplified2) print("dot^2 =", dot_sq) if dot_sq.is_rational: sqrt_val = sp.sqrt(dot_sq) print("sqrt(dot^2) =", sqrt_val) # But absolute value is sqrt(dot^2) if dot^2 positive. # Check sign. # Actually abs_dot = sqrt(dot^2) because dot^2 >=0. # So compute sqrt of rational. if dot_sq.is_Rational: num = dot_sq.p den = dot_sq.q # sqrt(num/den) may not be rational. # But if num/den is a perfect square rational? # Let's see. print("num, den:", num, den) # Simplify sqrt. sqrt_r = sp.sqrt(sp.Rational(num, den)) print("sqrt_r =", sqrt_r) if sqrt_r.is_rational: print("Rational sqrt:", sqrt_r) p = sqrt_r.q q = sqrt_r.p print("p+q =", p+q) else: # Then |PA·QB| is irrational, but problem says q/p rational. # So maybe we made mistake. pass [/PYTHON]

[THOUGHT] From the Python output, we have two possible Q satisfying conditions. One gives |PA·QB| = 196/25, with p=25, q=196 (coprime? 196/25 simplifies? 196 and 25 share no common factors, so yes coprime). Then p+q = 25+196 = 221. The other gives zero, but condition says |QB| > 0, so Q ≠ B. The solution Q = (5√2,0) is exactly point B (since B = (5√2,0)), which would make |QB| = 0, invalid. So we discard that. Thus the only valid Q gives |PA·QB| = 196/25, p+q = 221. [/THOUGHT] \boxed{221}