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\
[THOUGHT] Set A(a,b) on y=log_16(8x+2) so 8a+2=16^b. Let B(c,d) on y=4^{x-1}-1/2. Midpoint conditions give c=77/4-a and d=133/4-b. Reflection A'=(b,a) lies on line OB, so (b,a) is collinear with (c,d), i.e. ac=bd. Solve the resulting system and compute a*b as a reduced fraction. [/THOUGHT] [PYTHON] import sympy as sp
a,b = sp.symbols('a b', positive=True) c = sp.Rational(77,4) - a d = sp.Rational(133,4) - b
eq1 = sp.Eq(8a + 2, 16b) # A on log curve eq2 = sp.Eq(d, 4(c-1) - sp.Rational(1,2)) # B on exp curve eq3 = sp.Eq(ac, b*d) # collinearity condition
Use nsolve to find the solution in the first quadrant
sol = sp.nsolve([eq1, eq2, eq3], [a,b], [15.7, 1.7], tol=1e-30, maxsteps=100, prec=80) sol_a = sp.nsimplify(sol[0]) sol_b = sp.nsimplify(sol[1]) prod = sp.nsimplify(sol_asol_b) p, q = sp.fraction(sp.together(prod)) p, q = int(p), int(q) print("a,b =", sol_a, sol_b) print("ab =", prod) print("p+q =", p+q) [/PYTHON][THOUGHT] Set A(a,b) on y=log_16(8x+2) so 8a+2=16^b. Let B(c,d) on y=4^{x-1}-1/2. Midpoint conditions give c=77/4-a and d=133/4-b. Reflection A'=(b,a) lies on line OB, so (b,a) is collinear with (c,d), i.e. ac=bd. Solve the resulting system and compute a*b as a reduced fraction. [/THOUGHT] [PYTHON] import sympy as sp
a,b = sp.symbols('a b', positive=True) c = sp.Rational(77,4) - a d = sp.Rational(133,4) - b
eq1 = sp.Eq(8a + 2, 16b) # A on log curve eq2 = sp.Eq(d, 4(c-1) - sp.Rational(1,2)) # B on exp curve eq3 = sp.Eq(ac, b*d) # collinearity condition
sol = sp.nsolve([eq1, eq2, eq3], [a,b], [15.7, 1.7], tol=1e-30, maxsteps=100, prec=80) sol_a = sp.nsimplify(sol[0]) sol_b = sp.nsimplify(sol[1]) prod = sp.nsimplify(sol_asol_b) p, q = sp.fraction(sp.together(prod)) p, q = int(p), int(q) print("a,b =", sol_a, sol_b) print("ab =", prod) print("p+q =", p+q) [/PYTHON]
[THOUGHT] From the Python output, (a=\frac{63}{4}), (b=\frac{7}{4}), so (ab=\frac{441}{16}) and (p+q=457). \boxed{457}