Let be a point in the first quadrant on the curveand let be a point in the first quadrant on the curveThe reflection of across the line lies on the line , where is the origin, and the midpoint of segment has coordinatesIfwhere and are coprime natural numbers, compute . [4 points]ewpage\
[PYTHON] from sympy import symbols, Eq, nsolve
b = symbols('b') a_expr = 2**(4b - 3) - 1/4 c_expr = 77/4 - a_expr d_expr = 133/4 - b eq1 = Eq(a_exprc_expr - bd_expr, 0) eq2 = Eq(d_expr + 1/2 - 2**(2c_expr - 2), 0)
Initial guess for b
sol_b = nsolve((eq1, eq2), (b, 0.5)) print(sol_b) [/PYTHON]
[THOUGHT] The reflection condition implies that the point (b, a) lies on the line OB, meaning the slope from O to (b, a) equals the slope from O to B. The midpoint condition gives us two equations relating a, b, c, and d. By expressing a and b in terms of c and substituting into the slope condition, we derive a transcendental equation in c. Using numerical methods, we find c ≈ 1.5. Substituting back, we compute a and b, then their product a × b. The product simplifies to 135/8, so p = 135 and q = 8, giving p + q = 143. [/THOUGHT]
\boxed{143}