The focus of the ellipse x ^ 2 / 12 + y ^ 2 / 3 = 1 is F1, F2, and the point P is on the ellipse. If the midpoint of Pf1 is on the Y axis, then the ratio of Pf1 to PF2 is? Mathematical

The focus of the ellipse x ^ 2 / 12 + y ^ 2 / 3 = 1 is F1, F2, and the point P is on the ellipse. If the midpoint of Pf1 is on the Y axis, then the ratio of Pf1 to PF2 is? Mathematical

Answer: 7
From the title: a = 2, radical 3, B = radical 3, C = 3
Because the midpoint of Pf1 is on the y-axis, and the midpoint o of F1F2 is on the y-axis,
Therefore, PF2 is parallel to the y-axis, that is, PF2 ⊥ F1F2
Therefore, Pf1 & sup2; = PF2 & sup2; + F1F2 & sup2;, that is, Pf1 & sup2; - PF2 & sup2; = 6 & sup2; = 36 (1)
It is also defined by the ellipse: Pf1 + PF2 = 2A = 4 radical sign 3 -------- (2)
Solutions (1) and (2) are: Pf1 = 7 (radical 3) / 2, PF2 = (radical 3) / 2,
Therefore, the ratio of Pf1 to PF2 is 7

F 1 and F 2 are the left and right focal points of the ellipse x square / a square + y square / b square = 1 respectively. The distance from the right focal point to the upper vertex is 2. If a square = √ 6C, the elliptic equation can be solved

∵ b²＋c²=4=a²= √6c
∴ c²＝(4²/√6)²=8/3
∴ b²＝4=8/3=4/3
∴ a²＝4
The elliptic equation: X & # 178 / 4 + 3Y & # 178 / 4 = 1

The two focuses of the elliptic equation x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (greater than) are F1 and F2 respectively. The point P is on the ellipse C, and Pf1 is perpendicular to F1F2, Pf1 = 4
3, PF2 = 14 divided by 3, spherical elliptic C equation

PF1=4 /3,PF2=14/3
2a=4 /3+14/3=6
a=3
F1F2^2=(4/3)^2+(14/3)^2=4c^2
c^2=53/9
b^2=a^2-c^2=28/9
The elliptic C equation is x ^ 2 / 9 + 9y ^ 2 / 28 = 1