过椭圆x+2y^2=4的左焦点作倾斜角为π/3的斜线,求椭圆被直线截得的弦长

Question

过椭圆x+2y^2=4的左焦点作倾斜角为π/3的斜线,求椭圆被直线截得的弦长

Answer 1

x^2+2y^2=4
x^2/4+y^2/2=1
a^2=4,b^2=2,c^2=a^2-b^2=2
左焦点坐标:(-√2,0)
倾斜角为π/3,Kab=tanπ/3=√3
AB方程为：y= √3(x+√2)
代入：x^2+2y^2=4得：
x^2+2(√3(x+√2))^2=4
7x^2+12√2x+8=0
x1+x2=-12√2/7,x1x2=8/7
(x1-x2)^2=(x1+x2)^2-4x1x2=64/49
|AB|=√(1+k^2)(x1-x2)^2=√(1+3)*64/49=16/7
弦AB长为 16/7
很高兴为你解答,希望对你有所帮助,