Question

如圖,已知A,B,C是長軸長為4的橢圓上的三點,點A是長軸的一個端點,BC過橢圓中心O,且·=0,|BC|=2|AC|.?

(1)求橢圓方程;

(2)過點D(0,2)的直線l與橢圓相交于不同的兩點M,N且M在D,N之間,設=λ,求實數(shù)λ的取值范圍.?

Answer 1

解:(1)設A(2,0),則橢圓方程為=1.                                                  ?

∵O為橢圓中心,?

∴由對稱性知|OC|=|OB|.?

又∵·=0,∴AC⊥BC.?

又∵|BC|=2|AC|,∴|OC|=|AC|.?

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

∴點C的坐標為(1,1).?

∴點B的坐標為(-1,-1).?

將C的坐標(1,1)代入橢圓方程得b²=,                                                                  ?

則求得橢圓的方程為=1.                                                                          ?

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

設M(x₁,y₁),N(x₂,y₂),則x₁=,y₁=,                                                   ?

∵M,N在橢圓上,∴x₁²+3y₁²=4,x₂²+3y₂²=4.?

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

即3λy₂+2=2λy₂=.                                                                                    ?

∴-≤y₂≤.?

∴-≤≤.                                                                                ?

解得λ≥,?

∴實數(shù)λ的取值范圍是［,+∞).