Question

如图,已知A,B,C是长轴长为4的椭圆上的三点,点A是长轴的一个端点,BC过椭圆中心O,且·=0,|BC|=2|AC|.?

(1)求椭圆方程;

(2)过点D(0,2)的直线l与椭圆相交于不同的两点M,N且M在D,N之间,设=λ,求实数λ的取值范围.?

Answer 1

解:(1)设A(2,0),则椭圆方程为=1.                                                  ?

∵O为椭圆中心,?

∴由对称性知|OC|=|OB|.?

又∵·=0,∴AC⊥BC.?

又∵|BC|=2|AC|,∴|OC|=|AC|.?

∴△AOC为等腰直角三角形.                                                                                  ?

∴点C的坐标为(1,1).?

∴点B的坐标为(-1,-1).?

将C的坐标(1,1)代入椭圆方程得b²=,                                                                  ?

则求得椭圆的方程为=1.                                                                          ?

(2)∵M在D,N之间,∴λ＞0.                                                                              ?

设M(x₁,y₁),N(x₂,y₂),则x₁=,y₁=,                                                   ?

∵M,N在椭圆上,∴x₁²+3y₁²=4,x₂²+3y₂²=4.?

∴()²+()²=4λ²(x₂²+3y₂²)+12λy₂+12=4λ²+8λ+4,?

即3λy₂+2=2λy₂=.                                                                                    ?

∴-≤y₂≤.?

∴-≤≤.                                                                                ?

解得λ≥,?

∴实数λ的取值范围是［,+∞).