Question

证明双曲线试证明曲线y=k/x是双曲线,并求出其焦点与准线,而且是要求准线,不是渐进线,明白吗

Answer 1

将原坐标系旋转+45度,建立新直角坐标系x'oy'.

所以：

x^2+y^2=x'^2+y'^2

tana=y/x

tanb=y'/x'

a-b=45度

tan(a-b)=(y/x-y'/x')/(1+y/x*y'/x')=1

y/x-y'/x'=1+y/x*y'/x'

k/x^2-y'/x'=1+k/x^2*y'/x'

k/x^2(1-y'/x')=1+y'/x'

k/x^2=(1+y'/x')/(1-y'/x')=(x'+y')/(x'-y')

x^2=k(x'-y')/(x'+y')

y=k/x,xy=k

x'^2+y'^2=(x^2+k^2/x^2)=k(x'-y')/(x'+y')+k(x'+y')/(x'-y')

(x'^2+y'^2)(x'^2-y'^2)=k(x'2+y'^2-2x'y')+k(x'2+y'^2-2x'y')=2k(x'2+y'^2)

x'^2-y'^2=2k

所以是双曲线,

在x'oy'坐标系中：准线：x'=±根号(2k)/根号2=±根号k,焦点：(±2根号k,0)

转换成xoy坐标系：

准线：y-x=±根号(2k),

焦点：(根号k,根号k),(-根号k,-根号k).