椭圆x^2/a^2+y^2/b^2=1(a>b>0)满足a≤(√3)b,若离心率为e,求e^2+1/e^2的最小值.
来源:学生作业帮助网 编辑:作业帮 时间:2024/08/09 21:59:29
![椭圆x^2/a^2+y^2/b^2=1(a>b>0)满足a≤(√3)b,若离心率为e,求e^2+1/e^2的最小值.](/uploads/image/z/13867463-47-3.jpg?t=%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%E6%BB%A1%E8%B6%B3a%E2%89%A4%28%E2%88%9A3%29b%2C%E8%8B%A5%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BAe%2C%E6%B1%82e%5E2%2B1%2Fe%5E2%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC.)
x){d9mqFqFڕ@:)P#.@/mN|ԹDQ,c$K/t';v<ؔ
i$jy6
{lii~
ݟjV\Z`dm!-TH7 *cT+|hCi}8Xy1T@4FA15`,Ki,,#m "Bs5p7D[bPu@AcD3QRC}c';.3.iko3
Z $@lBh0(%C
}ڻPTlzZ`n(.H̳e
7t\
椭圆x^2/a^2+y^2/b^2=1(a>b>0)满足a≤(√3)b,若离心率为e,求e^2+1/e^2的最小值.
椭圆x^2/a^2+y^2/b^2=1(a>b>0)满足a≤(√3)b,若离心率为e,求e^2+1/e^2的最小值.
椭圆x^2/a^2+y^2/b^2=1(a>b>0)满足a≤(√3)b,若离心率为e,求e^2+1/e^2的最小值.
e=c/a,e²=c²/a²,又c²=a²-b²,
∴e²+1/e²=c²/a²+a²/c²
=(a²-b²)/a²+a²/(a²-b²)
=[2a²(a²-b²)+(b²)²]/[a²(a²-b²)]
=2+b²b²/[a²(a²-b²)]
=2+(b²/a²)[b²/(a²-b²)]
∵a≤(√3)b,∴a²≤3b²,
∴b²/a²≥1/3,且b²/(a²-b²)≥b²/(3b²-b²)=1/2
∴(b²/a²)[b²/(a²-b²)]≥(1/3)(1/2)=1/6
∴e²+1/e²=2+(b²/a²)[b²/(a²-b²)]≥2+1/6=13/6
即e²+1/e²的最小值是13/6.