如题所述
设F1ï¼-cï¼0ï¼ï¼F2ï¼cï¼0ï¼ï¼å
¶ä¸ c^2=a^2-b^2 ï¼
å¦è®¾ Pï¼xï¼yï¼æ¯æ¤åä¸ä»»ä¸ç¹ï¼
å PF1=ï¼-c-x ï¼-yï¼ï¼PF2=ï¼c-xï¼-yï¼ï¼
æä»¥ç± PF1*PF2=(-c-x)(c-x)+(-y)(-y)
=x^2+y^2-c^2
=x^2+b^2(1-x^2/a^2)-c^2
=c^2/a^2*x^2+b^2-c^2
=0
å¾ x^2=(c^2-b^2)a^2/c^2 ï¼
ç± 0<=x^2<=a^2 å¾ 0<=(c^2-b^2)a^2/c^2<=a^2 ï¼
å æ¤ b^2<=c^2 ï¼
ä¹å³ a^2-c^2<=c^2 ï¼
å æ¤è§£å¾ c^2/a^2>=1/2 ï¼
æ以 e>=â2/2 ï¼
å 为æ¤å离å¿çå°äº 1 ï¼
æ以å¾ææ±æ¤å离å¿ççèå´æ¯ [â2/2ï¼1ï¼ã
å¦è®¾ Pï¼xï¼yï¼æ¯æ¤åä¸ä»»ä¸ç¹ï¼
å PF1=ï¼-c-x ï¼-yï¼ï¼PF2=ï¼c-xï¼-yï¼ï¼
æä»¥ç± PF1*PF2=(-c-x)(c-x)+(-y)(-y)
=x^2+y^2-c^2
=x^2+b^2(1-x^2/a^2)-c^2
=c^2/a^2*x^2+b^2-c^2
=0
å¾ x^2=(c^2-b^2)a^2/c^2 ï¼
ç± 0<=x^2<=a^2 å¾ 0<=(c^2-b^2)a^2/c^2<=a^2 ï¼
å æ¤ b^2<=c^2 ï¼
ä¹å³ a^2-c^2<=c^2 ï¼
å æ¤è§£å¾ c^2/a^2>=1/2 ï¼
æ以 e>=â2/2 ï¼
å 为æ¤å离å¿çå°äº 1 ï¼
æ以å¾ææ±æ¤å离å¿ççèå´æ¯ [â2/2ï¼1ï¼ã
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