如图,双曲线的两顶点为A1,A2,虚轴两端点为 B1B2,两焦点为F1,F2.若以A1A2为直径的 圆内切于菱形F1B1F2B2,切点分别为A,B,C ,D.则 (Ⅰ)双曲线的离心率e=______;(Ⅱ)菱形F1B1F2B2的面积S1与矩形ABCD的
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/08 15:37:17
![如图,双曲线的两顶点为A1,A2,虚轴两端点为 B1B2,两焦点为F1,F2.若以A1A2为直径的 圆内切于菱形F1B1F2B2,切点分别为A,B,C ,D.则 (Ⅰ)双曲线的离心率e=______;(Ⅱ)菱形F1B1F2B2的面积S1与矩形ABCD的](/uploads/image/z/10192765-13-5.jpg?t=%E5%A6%82%E5%9B%BE%2C%E5%8F%8C%E6%9B%B2%E7%BA%BF%E7%9A%84%E4%B8%A4%E9%A1%B6%E7%82%B9%E4%B8%BAA1%2CA2%2C%E8%99%9A%E8%BD%B4%E4%B8%A4%E7%AB%AF%E7%82%B9%E4%B8%BA+B1B2%2C%E4%B8%A4%E7%84%A6%E7%82%B9%E4%B8%BAF1%2CF2.%E8%8B%A5%E4%BB%A5A1A2%E4%B8%BA%E7%9B%B4%E5%BE%84%E7%9A%84+%E5%9C%86%E5%86%85%E5%88%87%E4%BA%8E%E8%8F%B1%E5%BD%A2F1B1F2B2%2C%E5%88%87%E7%82%B9%E5%88%86%E5%88%AB%E4%B8%BAA%2CB%2CC+%2CD.%E5%88%99+%EF%BC%88%E2%85%A0%EF%BC%89%E5%8F%8C%E6%9B%B2%E7%BA%BF%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87e%3D______%EF%BC%9B%EF%BC%88%E2%85%A1%EF%BC%89%E8%8F%B1%E5%BD%A2F1B1F2B2%E7%9A%84%E9%9D%A2%E7%A7%AFS1%E4%B8%8E%E7%9F%A9%E5%BD%A2ABCD%E7%9A%84)
如图,双曲线的两顶点为A1,A2,虚轴两端点为 B1B2,两焦点为F1,F2.若以A1A2为直径的 圆内切于菱形F1B1F2B2,切点分别为A,B,C ,D.则 (Ⅰ)双曲线的离心率e=______;(Ⅱ)菱形F1B1F2B2的面积S1与矩形ABCD的
如图,双曲线的两顶点为A1,A2,虚轴两端点为 B1B2,两焦点为F1,F2.若以A1A2为直径的 圆内切于菱形F1B1F2B2,切点分别为A,B,C ,D.则 (Ⅰ)双曲线的离心率e=______;
(Ⅱ)菱形F1B1F2B2的面积S1与矩形ABCD的面 积S2的比值__________.
如图,双曲线的两顶点为A1,A2,虚轴两端点为 B1B2,两焦点为F1,F2.若以A1A2为直径的 圆内切于菱形F1B1F2B2,切点分别为A,B,C ,D.则 (Ⅰ)双曲线的离心率e=______;(Ⅱ)菱形F1B1F2B2的面积S1与矩形ABCD的
双曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的两顶点为A1(-a,0),A2(a,0),虚轴两端点为 B1(0,-b),B2(0,b),两焦点为F1(-c,0),F2(c,0).
(I)若以A1A2为直径的 圆内切于菱形F1B1F2B2,切点分别为A,B,C ,D,则
a√(c^2+b^2)=bc,
平方得a^2(c^2+b^2)=b^2c^2,b^2=c^2-a^2,
∴a^2(2c^2-a^2)=c^2(c^2-a^2),e=c/a,
∴2e^2-1=e^2(e^2-1),
∴e^4-3e^2+1=0,e>1,
∴e^2=(3+√5)/2,
e=(√5+1)/2.
(II)设点A在第一象限,B在第四象限,tanAOF2=tanOB2F2=c/b,
由万能公式,sinAOB=(2c/b)/[1+(c/b)^2]=2bc/(b^2+c^2)
∴S1/S2=2bc/[2a^2*2bc/(b^2+c^2)]
=(b^2+c^2)/a^2
=(2c^2-a^2)/a^2
=2e^2-1
=2+√5.