Given that F1 and F2 are two focal points of the ellipse X & sup2 / 100 + Y & sup2 / 64 = 1, and P is a point on the ellipse, find the maximum value of Pf1 × PF2

Given that F1 and F2 are two focal points of the ellipse X & sup2 / 100 + Y & sup2 / 64 = 1, and P is a point on the ellipse, find the maximum value of Pf1 × PF2

a2=100
a=10
Let Pf1 = m, PF2 = n
Then M + n = 2A = 20
m>0,n>0
So m + n ≥ 2 √ Mn
2√mn≤20
mn≤100
So the maximum value is 100

It is known that F1 and F2 are the two focal points of the ellipse X225 + Y29 = 1, and point P is a moving point on the ellipse, then the minimum value of | Pf1 | · | PF2 | is______ ．

Let | Pf1 | = x, ∵ Pf1 | + | PF2 | = 2A = 10, ∵ PF2 | = 10-x | Pf1 | ·| PF2 | = x (10-x) = - x2 + 10x = - (X-5) 2 + 25 ∵ in the ellipse, a = 5, B = 3, C = 4, ∵ 1 ≤ x ≤ 9 ∵ function y = - x2 + 10x monotonically increases on [1,5] and decreases monotonically on [5,9]; when x = 1 or 9, y = - x2 + 1

P is the point on the ellipse x ^ 2 / 25 + y ^ 2 / 16 = 1, and the left and right focus are F1 and F2. Find the maximum value of (Pf1) (PF2)

The maximum value of Pf1 * PF2 is 25 and the minimum value is 16