A\P\B是双曲线上三点,x2/a2-y2/b2=1(a>0,b>0),且A\B连线过原点,PA与PB斜率的乘积=5/3,球双曲线离心率
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![A\P\B是双曲线上三点,x2/a2-y2/b2=1(a>0,b>0),且A\B连线过原点,PA与PB斜率的乘积=5/3,球双曲线离心率](/uploads/image/z/6956918-62-8.jpg?t=A%5CP%5CB%E6%98%AF%E5%8F%8C%E6%9B%B2%E7%BA%BF%E4%B8%8A%E4%B8%89%E7%82%B9%2Cx2%2Fa2-y2%2Fb2%3D1%28a%3E0%2Cb%3E0%29%2C%E4%B8%94A%5CB%E8%BF%9E%E7%BA%BF%E8%BF%87%E5%8E%9F%E7%82%B9%2CPA%E4%B8%8EPB%E6%96%9C%E7%8E%87%E7%9A%84%E4%B9%98%E7%A7%AF%3D5%2F3%2C%E7%90%83%E5%8F%8C%E6%9B%B2%E7%BA%BF%E7%A6%BB%E5%BF%83%E7%8E%87)
A\P\B是双曲线上三点,x2/a2-y2/b2=1(a>0,b>0),且A\B连线过原点,PA与PB斜率的乘积=5/3,球双曲线离心率
A\P\B是双曲线上三点,x2/a2-y2/b2=1(a>0,b>0),且A\B连线过原点,PA与PB斜率的乘积=5/3,球双曲线离心率
A\P\B是双曲线上三点,x2/a2-y2/b2=1(a>0,b>0),且A\B连线过原点,PA与PB斜率的乘积=5/3,球双曲线离心率
设:A(m,n)、B(-m,-n)、P(p,q)
PA斜率是:[q-n]/[p-m];PB斜率是:[q+n]/[p+n],则:
[(q-n)/(p-m)]×[(q+n)/(p+m)]=5/3
[q²-n²]/[p²-m²]=5/3 ---------------------(1)
因:
m²/a²-n²/b²=1、p²/a²-q²/b²=1
两式相减,得:
(m²-p²)/a²-(n²-q²)/b²=0
则:
(q²-n²)/(p²-m²)=b²/a²=5/3
5a²=3b²=3(c²-a²)
3c²=8a²
e=c/a=√(8/3)=(2√6)/3