From an end point of the ellipse minor axis to see the two focal points, the angle of view is 120 to calculate the eccentricity

From an end point of the ellipse minor axis to see the two focal points, the angle of view is 120 to calculate the eccentricity

Let the elliptic equation be: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1, (a > b > 0), and the vertex of the minor axis be a (0, b),
The focal coordinates are F1 (- C, 0), F2 (C, 0),

The symmetry axis of the ellipse is the coordinate axis. One end point of the minor axis and two focal points form three vertices of an equilateral triangle. The shortest distance from the focal point to the point on the ellipse
For the root sign three, find the standard equation of the ellipse

The ellipse parameters a, B and C can be obtained from the problem
a-c=√3 ①
b=√3c ②
a=2c ③
The simultaneous solutions of (1) and (3) are: a = 2 √ 3, B = 3, and the elliptic equation is:
x²/12+y²/9=1
Or X & sup2 / 9 + Y & sup2 / 12 = 1

The standard equation of hyperbola with the focus of ellipse & sup2; X / 169 + Y & sup2; 144 = 1 is

Let the vertex of the ellipse be (± 13,0) and the focus be (± 5,0)
Then the focus of the hyperbola is (± 13,0) and the vertex is (± 5,0)
Let the hyperbolic equation be x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1, then a = 5, B ^ 2 = 13 ^ 2-5 ^ 2 = 144,
So the hyperbolic equation is x ^ 2 / 25-y ^ 2 / 144 = 1