Hướng dẫn giải

a) F₁M2 = [x – (– c)]2 + (y – 0)2 = (x + c)2 + y2 = x2 +2cx + c2 + y2;

F₂M2 = (x – c)2 +(y – 0)2= x2 -2cx + c2 + y2

F₁M2 – F₂M2 = (x2 +2cx + c2 + y2) – (x2 -2cx + c2 + y2) = 4cx.

b) Ta có: MF₁2 – MF₂2 = 4cx => (MF₁ + MF₂)(MF₁ – MF₂) = 4cx => (MF₁ + MF₂)(–2a) = 4cx

=> MF₁ + MF₂ = 4cx2a = –2cax. Khi đó:

(MF₁ + MF₂) + (MF₁ – MF₂) = –2ca + (–2a) => 2MF₁ = –

2ca – 2a

=> MF₁ = −(cax+a)=−a−cax.

(MF₁ + MF₂) – (MF₁ – MF₂) = –2ca – (–2a) => 2MF₂ = -2ca + 2a

=> MF₂ = a –c/a x.

c) Ta có: MF₁2 – MF₂2 = 4cx => (MF₁ + MF₂)(MF₁ – MF₂) = 4cx => (MF₁ + MF₂)2a = 4cx

=> MF₁ + MF₂ = 4cx2a = 2cax. Khi đó:

(MF₁ + MF₂) + (MF₁ – MF₂) =2ca + 2a => 2MF₁ =2ca + 2a

=> MF₁ = a + cax.

(MF₁ + MF₂) – (MF₁ – MF₂) =2ca – 2a => 2MF₂ =2ca – 2a

=> MF₂ = – a +cax.

Cho hypebol x^2/a^2 - y^2/b^2 =1. Chứng tỏ rằng c/a >1