Question

双曲线提问若双曲线方程x²-y²/3=1,过双曲线的左焦点,倾斜角为π/6的直线方程与双曲线相较于A

Answer 1

a^2=1

b^2=3

c^2=4

所以左焦点F1(-2,0)

k=tanπ/6=√3/3

y=√3/3(x+2)

代入

3x^2-(1/3)*(x+2)^2=3

9x^2-x^2-4x-4=9

8x^2-4x-13=0

x1+x2=1/2,x1x2=-13/8

(x1-x2)^2=(x1+x2)^2-4x1x2=1/4+13/2=27/4

(y1-y2)^2=[√3/3(x1+2)-√3/3(x2+2)]^2=[√3/3(x1-x2)]^2=1/3*27/4=9/4

所以AB^2=(x1-x2)^2+(y1-y2)^2=9

|AB|=3