If ellipse (X & # 178 / M) + Y & # 178; = 1 (M > 1) and hyperbola (X & # 178 / N) - Y & # 178; = 1 (n > 0) have the same focus F1 and F2 If P is an intersection of two curves, what is the value of vector Pf1 * vector PF2

If ellipse (X & # 178 / M) + Y & # 178; = 1 (M > 1) and hyperbola (X & # 178 / N) - Y & # 178; = 1 (n > 0) have the same focus F1 and F2 If P is an intersection of two curves, what is the value of vector Pf1 * vector PF2

Ellipse (X & # 178 / M) + Y & # 178; = 1 (M > 1)
A & # 178; = m, B = 1, C = √ (A & # 178; - B & # 178;) = √ (M - 1), focus F1 (- √ (M - 1), 0), F2 (√ (M - 1), 0)
Hyperbola (X & # 178 / N) - Y & # 178; = 1 (n > 0)
a² = n,b = 1,c = √(a² + b²) = √(n + 1)
They have the same focus, √ (M - 1) = √ (n + 1)
m = n + 2 (1)
Add the two, X & # 178; = n (n + 2) / (n + 1)
y² = 1/(n + 1)
PF1 = (-√(n + 1) - x,y),PF2 = (√(n + 1) - x,y)
Vector Pf1 * vector PF2 = (- x) &# 178; - (n + 1) + Y & # 178;
= n(n + 2)/(n+1) - (n+1) + 1/(n+1)
= 0