令P(x1,y1),Q(x2,y2).ç´çº¿PQæ¹ç¨ä¸º:y=kx+m(å
¶ä¸k=(y2-y1)/(x2-x1)) â 代å
¥x^2/4+y^2=1â¡å¹¶æ´çå¾:
(1+4k^2)x^2+8kmx+4m^2-4=0. â¢
ä¾é¢ææÎ=16(1+4k^2-m^2)>0. x1+x2=-(8km)/ (1+4k^2).
| x2-x1|=âÎ]/(1+4k^2), | y2-y1|=|k( x2-x1)|= |k|âÎ]/(1+4k^2),
x1x2=4(m^2-1) /(1+4k^2),
y1y2= (kx1+m)( kx2+m)=k^2x1x2+km(x1+x2)+m^2
=4 k^2(m^2-1) /(1+4k^2) - (8k^2m^2)/ (1+4k^2) + m^2
OP,PQ,OQæçæçæ¯æ°å,åæ(y2-y1)^2/(x2-x1)^2=y1y2/x1x2. â£
å³k^2=[4 k^2(m^2-1) /(1+4k^2)-(8k^2m^2)/ (1+4k^2)+m^2]/ [4(m^2-1) /(1+4k^2)]
æ´çå¾:k^2=1/4.
|PQ|=[â(1+k^2)âÎ]/(1+4k^2)= [4â(1+k^2)â(1+4k^2-m^2)]/(1+4k^2) â¤
Oå°ç´çº¿PQçè·ç¦»d=|m|/(â(1+k^2) â¥
â³OPQé¢ç§¯=|PQ|*d/2=[4â(1+k^2)â(1+4k^2-m^2)]/(1+4k^2) *|m|/(â(1+k^2)/2 =2|m|â(1+4k^2-m^2)/(1+4k^2) =2|m|â(1+1-m^2)/(1+1)
=|m|â(2-m^2) â¦
æ¾ç¶â³OPQé¢ç§¯å¨|m|=1æ¶åå¾æ大å¼1,æå°å¼ä¸º0,æ
åå¼èå´(0,1].
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