In the coordinate plane, let be a segment of length 10$$\sqrt{2}$$$, and consider the circle with diameter ABPQ(\overrightarrow{\text{PA}} + \overrightarrow{\text{PB}}) \cdot (\overrightarrow{\text{PQ}} + \overrightarrow{\text{PB}}) = 2\left|\overrightarrow{\text{PQ}}\right|^2\left|\overrightarrow{\text{PB}}\right| = 14\left|\overrightarrow{\text{QB}}\right| > 0\left|\overrightarrow{\text{PA}} \cdot \overrightarrow{\text{QB}}\right| = \dfrac{q}{p}pqp + q$. [4 points]
[THOUGHT] I'll set up symbolic coordinates for points A, B, P, Q, encode the circle constraints and the given vector/dot-product condition, then solve the resulting system with sympy. After obtaining all valid solutions, I'll compute the value(s) of |PA · QB| and identify the rational value q/p, then output p+q. [/THOUGHT] [PYTHON] from sympy import symbols, Eq, sqrt, solve, Matrix, simplify, nsimplify
Define symbols
px, py, qx, qy = symbols('px py qx qy', real=True) s2 = sqrt(2)
Constants and fixed points
A = Matrix([-5s2, 0]) B = Matrix([ 5s2, 0]) P = Matrix([px, py]) Q = Matrix([qx, qy])
Circle radius^2
r2 = (5*s2)**2 # should be 50
Vectors
PA = A - P PB = B - P PQ = Q - P QB = B - Q
Equations
eq_circle_P = Eq(px2 + py2, r2) eq_circle_Q = Eq(qx2 + qy2, r2) eq_PB_len = Eq((px-5*s2)2 + py2, 14**2)
lhs_vec = (PA + PB).dot(PQ + PB) rhs_vec = 2 * (PQ.dot(PQ)) eq_dot = Eq(lhs_vec, rhs_vec)
eq_circle_P, eq_circle_Q, eq_PB_len, eq_dot print(eq_circle_P) print(eq_circle_Q) print(eq_PB_len) print(eq_dot)
Solve system
solutions = solve([eq_circle_P, eq_circle_Q, eq_PB_len, eq_dot], [px, py, qx, qy], dict=True)
len(solutions)[THOUGHT] I need to see the solutions from Python to determine |PA · QB| and then express it as q/p. [/THOUGHT]
[THOUGHT] The previous code failed due to a Unicode middle-dot character inside a comment. I'll rewrite the code without any such characters, re-solve the system, then compute |PA · QB| from the valid solution(s). Finally I'll convert that value to q/p and compute p+q. [/THOUGHT] [PYTHON] from sympy import symbols, Eq, sqrt, solve, Matrix, simplify, nsimplify, Rational
Define symbols
px, py, qx, qy = symbols('px py qx qy', real=True) s2 = sqrt(2)
Fixed points A, B (AB has length 10*sqrt(2))
A = Matrix([-5s2, 0]) B = Matrix([ 5s2, 0])
P = Matrix([px, py]) Q = Matrix([qx, qy])
Circle radius^2 (circle with diameter AB, center at origin)
r2 = (5*s2)**2 # should be 50
Vectors
PA = A - P PB = B - P PQ = Q - P QB = B - Q
Equations:
1) P on circle
eq_circle_P = Eq(px2 + py2, r2)
2) Q on circle
eq_circle_Q = Eq(qx2 + qy2, r2)
3) |PB| = 14
eq_PB_len = Eq((px-5*s2)2 + py2, 14**2)
4) (PA + PB) dot (PQ + PB) = 2|PQ|^2
lhs_vec = (PA + PB).dot(PQ + PB) rhs_vec = 2 * (PQ.dot(PQ)) eq_dot = Eq(lhs_vec, rhs_vec)
print(eq_circle_P) print(eq_circle_Q) print(eq_PB_len) print(eq_dot)
solutions = solve( [eq_circle_P, eq_circle_Q, eq_PB_len, eq_dot], [px, py, qx, qy], dict=True )
print("Number of solutions:", len(solutions)) for sol in solutions: print("Solution:", sol) # Reconstruct vectors for this solution P_sol = Matrix([sol[px], sol[py]]) Q_sol = Matrix([sol[qx], sol[qy]])
PA_sol = A - P_sol
QB_sol = B - Q_sol
dot_val = simplify(PA_sol.dot(QB_sol))
abs_dot = simplify(abs(dot_val))
print("PA·QB =", dot_val, " |PA·QB| =", abs_dot)
From the absolute dot values, identify the rational q/p and compute p+q
We'll collect distinct abs values and attempt nsimplify as a Rational.
abs_values = [] for sol in solutions: P_sol = Matrix([sol[px], sol[py]]) Q_sol = Matrix([sol[qx], sol[qy]]) PA_sol = A - P_sol QB_sol = B - Q_sol dot_val = simplify(PA_sol.dot(QB_sol)) abs_dot = simplify(abs(dot_val)) abs_values.append(abs_dot)
print("Absolute dot product values:", abs_values)
Simplify each to a rational if possible
rats = [] for v in abs_values: r = nsimplify(v, [sqrt(2)]) rats.append(r) print("nsimplify:", v, "->", r)
Assume all valid solutions give same |PA·QB|
if rats: r0 = rats[0] r0_qp = Rational(r0) p = r0_qp.q q = r0_qp.p print("Chosen |PA·QB| =", r0_qp, " = q/p with q =", q, "p =", p, "p+q =", p+q) [/PYTHON]
[THOUGHT] From the valid solutions with |QB|>0, we get (|\overrightarrow{PA}\cdot\overrightarrow{QB}| = \frac{196}{25}), so (q=196), (p=25), and (p+q=221). [ \boxed{221} ]