If the point F is the left focus of the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0), the point F is the tangent of the circle x ^ 2 + y ^ 2 = B ^ 2, intersects the ellipse at the point P, and the tangent point q is the midpoint of the line FP, then the eccentricity of the ellipse is 0

If the point F is the left focus of the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0), the point F is the tangent of the circle x ^ 2 + y ^ 2 = B ^ 2, intersects the ellipse at the point P, and the tangent point q is the midpoint of the line FP, then the eccentricity of the ellipse is 0

e=v5/3.
The reasons are as follows:
Let the right focus be e
Pf = 2V (C ^ 2-B ^ 2)
Connect Po
Because Po = fo = OE
So the three points of PFE are the three points on the circle with o as the center
PFE is a right triangle
PF＾2＋PE＾2＝FE＾2
PE = 2B
Pf + PE = 2A
A = 3 / 2B
c＝v5／2b
So C / a = V5 / 3
Hope you are satisfied!