It is known that the left focus of the ellipse x ^ 2 + y ^ 2 / b ^ 2 = 1 (b belongs to (0,1)) is f, the left and right vertices are AC, the upper vertex is B, and the circle P is made through three points of FBC, where the coordinates of the center P are (m, n). Let f, B and C be (- C, 0), (0, b), (1,0) respectively. If the eccentricity of the ellipse e = 2 / 3, the equation of the circle P is obtained

It is known that the left focus of the ellipse x ^ 2 + y ^ 2 / b ^ 2 = 1 (b belongs to (0,1)) is f, the left and right vertices are AC, the upper vertex is B, and the circle P is made through three points of FBC, where the coordinates of the center P are (m, n). Let f, B and C be (- C, 0), (0, b), (1,0) respectively. If the eccentricity of the ellipse e = 2 / 3, the equation of the circle P is obtained

Center P (m, n)
m=(2-√3)/4,n=(1-2√3)/4
Radius r = √ 20

In the elliptic equation, if the circle with the diameter of two focal points just passes through the two vertices of the minor axis of the ellipse, then the eccentricity of the ellipse is?

Because the circle with the diameter of the two focal points just passes through the two vertices of the minor axis of the ellipse, the origin is the center of the circle, so 2C = 2B
Then C = B, B ^ 2 = a ^ 2-C ^ 2, C ^ 2 / A ^ 2 = a ^ 2, C ^ 2 / A ^ 2 = 1 / 2, so e = two-thirds root 2