Question

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

(1)求橢圓方程;

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

Answer 1

解:(1)設(shè)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(diǎn)C的坐標(biāo)為(1,1).?

∴點(diǎn)B的坐標(biāo)為(-1,-1).?

將C的坐標(biāo)(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(shè)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(shí)數(shù)λ的取值范圍是［,+∞).