双曲线C:x2/a2-y2/b2=1的右焦点为F,过F且斜率为根号3的直线交C于A、B两点,若AF=4FB,则C的离心率为?
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![双曲线C:x2/a2-y2/b2=1的右焦点为F,过F且斜率为根号3的直线交C于A、B两点,若AF=4FB,则C的离心率为?](/uploads/image/z/1670543-71-3.jpg?t=%E5%8F%8C%E6%9B%B2%E7%BA%BFC%EF%BC%9Ax2%2Fa2-y2%2Fb2%3D1%E7%9A%84%E5%8F%B3%E7%84%A6%E7%82%B9%E4%B8%BAF%2C%E8%BF%87F%E4%B8%94%E6%96%9C%E7%8E%87%E4%B8%BA%E6%A0%B9%E5%8F%B73%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%BA%A4C%E4%BA%8EA%E3%80%81B%E4%B8%A4%E7%82%B9%2C%E8%8B%A5AF%3D4FB%2C%E5%88%99C%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BA%3F)
双曲线C:x2/a2-y2/b2=1的右焦点为F,过F且斜率为根号3的直线交C于A、B两点,若AF=4FB,则C的离心率为?
双曲线C:x2/a2-y2/b2=1的右焦点为F,过F且斜率为根号3的直线交C于A、B两点,若AF=4FB,则C的离心率为?
双曲线C:x2/a2-y2/b2=1的右焦点为F,过F且斜率为根号3的直线交C于A、B两点,若AF=4FB,则C的离心率为?
设AF=4m,BF=m.过A,B分别做准线的垂线,垂足为A1,B1.有双曲线定义得,
AA1=4m/e.BB1=m/e.过B做BD垂直于AA1垂足D.三角形ABD中,角ABD=30°,AD=0.5AB.即3m/e=0.5×5m.解得e=6/5.
设A(c+4tcos60°,4tsin60°),B(c-tcos60°,-tsin60°),
A,B都在双曲线上,
∴(c+4tcos60°)^2/a^2-(4tsin60°)^2/b^2=1,①
(c-tcos60°)^2/a^2-(-tsin60°)^2/b^2=1,②,
②*16-①,16(c-tcos60°)^2-(c+4tcos60°)^2=15a^2,<...
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设A(c+4tcos60°,4tsin60°),B(c-tcos60°,-tsin60°),
A,B都在双曲线上,
∴(c+4tcos60°)^2/a^2-(4tsin60°)^2/b^2=1,①
(c-tcos60°)^2/a^2-(-tsin60°)^2/b^2=1,②,
②*16-①,16(c-tcos60°)^2-(c+4tcos60°)^2=15a^2,
16c^2-16ct+4t^2-(c^2+4ct+4t^2)=15a^2,
15c^2-20ct=15a^2,t=3b^2/(4c),
代入②,[c-3b^2/(8c)]^2/a^2-[3√3b^2/(8c)]^2/b^2=1,
(5c^2+3a^2)^2/a^2-27b^2=64c^2,e=c/a,
(5e^2+3)^2-27(e^2-1)=64,
25e^4+3e^2-28=0,e>1,本题无解.
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