Problem Analysis #geometry_29

Recorded Python call:

import sympy as sp

# Define symbols
p, y = sp.symbols('p y', positive=True, real=True)

# Equation from parabola and area:
# y^2 = 4p*x => x = y^2/(4p)
# Area: (1/2)*(x+p)*y = 2p+12
# => (y^2/(4p) + p)*y = 4p+24
# => y^3/(4p) + p*y = 4p+24
eq_area = sp.Eq(y**3/(4*p) + p*y, 4*p + 24)
# Multiply by 4p
eq_area = sp.Eq(y**3 + 4*p**2*y, 16*p**2 + 96*p)

# Try integer solutions for p
solutions = []
for p_val in range(1, 30):
    # Solve cubic for y
    eq = y**3 + 4*p_val**2*y - 16*p_val**2 - 96*p_val
    # Find real roots
    roots = sp.nroots(eq)
    for root in roots:
        if abs(sp.im(root)) < 1e-10 and sp.re(root) > 0:
            y_val = float(sp.re(root))
            x_val = y_val**2 / (4*p_val)
            # Check if x_val is positive
            if x_val > 0:
                solutions.append((p_val, y_val, x_val))

print("Integer p solutions:")
for sol in solutions:
    print(f"p={sol[0]}, y={sol[1]:.6f}, x={sol[2]:.6f}")

Recorded Python call:

import sympy as sp

# Define the values for p=9
p_val = 9
y_val = 6
x_val = 1

# Foci
b = 13*sp.sqrt(3)/2 - 4
c = -4 - 13*sp.sqrt(3)/2

# Points
F = (p_val, 0)
A = (x_val, y_val)
H = (-p_val, y_val)
B = (b, 0)
C = (c, 0)

# Distance function
def dist(P, Q):
    return sp.sqrt((P[0]-Q[0])**2 + (P[1]-Q[1])**2)

# Sums
sum_F = dist(F, B) + dist(F, C)
sum_A = dist(A, B) + dist(A, C)
sum_H = dist(H, B) + dist(H, C)

print("Sum from F:", sp.simplify(sum_F))
print("Sum from A:", sp.simplify(sum_A))
print("Sum from H:", sp.simplify(sum_H))

# Check if they are equal
print("F == A?", sp.simplify(sum_F - sum_A))
print("F == H?", sp.simplify(sum_F - sum_H))

# Perimeter check
AH = dist(A, H)
HB = dist(H, B)
AB = dist(A, B)
perimeter = AH + HB + AB
print("AH:", AH)
print("HB:", sp.simplify(HB))
print("AB:", sp.simplify(AB))
print("Perimeter:", sp.simplify(perimeter))

# Area
area = sp.Rational(1,2) * AH * y_val
print("Area:", area)

# HF
HF = dist(H, F)
print("HF:", sp.simplify(HF))
print("k^2:", sp.simplify(HF**2))

Recorded Python call:

# Verify that HB + AB = 26
HB_plus_AB = sp.sqrt(751 - 260*sp.sqrt(3))/2 + sp.sqrt(751 + 260*sp.sqrt(3))/2
print("HB + AB:", sp.simplify(HB_plus_AB))
print("Perimeter = AH + HB + AB:", sp.simplify(10 + HB_plus_AB))

Recorded Python call:

import sympy as sp
HB_plus_AB = sp.sqrt(751 - 260*sp.sqrt(3))/2 + sp.sqrt(751 + 260*sp.sqrt(3))/2
print("HB + AB:", sp.simplify(HB_plus_AB))
print("Perimeter:", sp.simplify(10 + HB_plus_AB))

Recorded Python call:

import sympy as sp
expr = sp.sqrt(751 - 260*sp.sqrt(3))/2 + sp.sqrt(751 + 260*sp.sqrt(3))/2
print("Numerical value:", expr.evalf())
print("Simplified:", sp.simplify(expr))

Recorded Python call:

# Let's also verify p=3 case to be sure it fails
import sympy as sp
p_val = 3
y_val = 6
x_val = 3
# From perimeter condition: HB + AB = 27 - x = 24
# Solve for b: sqrt((b+3)^2+36) + sqrt((b-3)^2+36) = 24
b = sp.symbols('b', real=True)
eq = sp.Eq(sp.sqrt((b+3)**2 + 36) + sp.sqrt((b-3)**2 + 36), 24)
# Solve
sol_b = sp.solve(eq, b)
print("Solutions for b:", sol_b)
# Check if any real solution exists
for s in sol_b:
    print(s, "approx:", s.evalf())

최종 풀이

주어진 조건 정리:

• 포물선: 초점 $F(p,0)$, 준선 $x=-p$ → 방정식 $y^2 = 4px$
• 점 $A$가 포물선 위, 제1사분면에 있음
• $H$는 $A$에서 준선에 내린 수선의 발 → $H(-p, y)$
• $x$좌표 양수인 타원의 초점 $B$
• 삼각형 $AHB$의 둘레 $= p+27$, 넓이 $= 2p+12$

계산:

1. 넓이 조건:

2. 둘레 조건:

3. 타원의 성질: $A$와 $H$가 타원 위에 있으므로, 타원 중심은 $AH$의 중점 $\left(\frac{x-p}{2}, y\right)$이고, 타원의 장축 방정식은 $AB + AH$의 길이가 일정해야 함.

4. 정수해 탐색:

• $p=3, x=3, y=6$인 경우: 타원 조건 검증 시 조건 불충족
• $p=9, x=1, y=6$인 경우: 모든 조건 만족

검증:

• 포물선 방정식: $y^2 = 36 = 4 \cdot 9 \cdot 1 = 36$ ✓
• 넓이: $\frac{1}{2} \cdot 10 \cdot 6 = 30 = 2(9)+12$ ✓
• 둘레: $10 + HB + AB = 36 = 9+27$ ✓

타원의 구성:

• 중심: $(0, 0)$
• 장축의 길이: $HB + AB = 26$ → $2a = 26$ → $a = 13$
• 초점 거리: $c = 4$ (중심에서 $B$까지)
• 준선의 길이: $a^2 - c^2 = 169 - 16 = 153$

$k$ 값 계산:

결과: