Question

过椭圆x^2+2y^2=4的左焦点作倾斜角为30度的直线，则弦长为

Answer 1

x�0�5+2y�0�5=4

即 x�0�5/4+y�0�5/2=1

a�0�5=4,b�0�5=2,c�0�5=a�0�5-b�0�5=2

左焦点坐标:(-√2,0)

倾斜角为30°,直线AB斜率 k=tan30°=√3/3

AB方程为：y=√3/3(x+√2)

代入：x�0�5+2y�0�5=4得：

x�0�5+2[√3/3(x+√2)]�0�5=4

5x�0�5+4√2x-8=0

x1+x2=-4/5√2 ,x1x2=-8/5

(x1-x2)�0�5=(x1+x2)�0�5-4x1x2=72/25

|AB|=√[(1+k�0�5)(x1-x2)�0�5]=√[(1+1/3)*72/25]=4/5√6

弦AB长为 4/5√6