Ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) and hyperbola x ^ 2 / M-Y ^ 2 / N = 1 (m, n > 0) have common focus, F1, F2, P are their common points Ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) and hyperbola x ^ 2 / M-Y ^ 2 / N = 1 (m, n > 0) have common focus F1, F2 and P are their common points (1) Using B and N to express cos ∠ f1pf2 (2) Let s △ f1pf2 = f (B, n)

Ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) and hyperbola x ^ 2 / M-Y ^ 2 / N = 1 (m, n > 0) have common focus, F1, F2, P are their common points Ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) and hyperbola x ^ 2 / M-Y ^ 2 / N = 1 (m, n > 0) have common focus F1, F2 and P are their common points (1) Using B and N to express cos ∠ f1pf2 (2) Let s △ f1pf2 = f (B, n)

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/PF1/+/PF2/=2a /F1F2/=2c / /PF1/-/PF2/ /=2√m
(/PF1/+/PF2/)²=4a²
(/PF1/-/PF2/)²=4m
Subtract the above formula to / Pf1 / × / PF2 / = A & # 178; - M
And because
a²-b²=c² m+n=c²
So / Pf1 / × / PF2 / = n + B & # 178;
/F1F2/²=4c²=4m+4n
So / Pf1 / &# 178; + / PF2 / &# 178; = (/ Pf1 / + / PF2 /) &# 178; - 2 / Pf1 / × / PF2/
=4a²-2（n+b²)
=4m+2n+2b²
So cos ∠ f1pf2 = (/ Pf1 / &# 178; + / PF2 / &# 178; - / F1F2 / &# 178;) divided by (2 / Pf1 / × / PF2 /)
=(4m + 2n + 2B & # 178; - 4m-4n) divided by [2 × (n + B & # 178;)]
= (- N + B & # 178;) divided by (n + B & # 178;)
（2）S²＝1/4×/PF1/²/PF2/²sin² ∠F1PF2
=1/4×（n+b²)²×(1-cos²∠F1PF2)
=1/4 ×[(n+b²)²-(-n+b²)²]
=1/4×(n+b²-n+b²)(n+b²+n-b²)
=nb²
So s = √ NB & # 178;
That is, f (B, n) = √ NB & # 178;