Question

已知雙曲線(xiàn)C:-y²=1.

(1)求雙曲線(xiàn)C的漸近線(xiàn)方程;

(2)已知點(diǎn)M的坐標(biāo)為(0,1),設(shè)P是雙曲線(xiàn)C上的點(diǎn),Q是點(diǎn)P關(guān)于原點(diǎn)的對(duì)稱(chēng)點(diǎn).記λ= ·,求λ的取值范圍;

(3)已知點(diǎn)D、E、M的坐標(biāo)分別為(-2,-1)、(2,-1)、(0,1),P為雙曲線(xiàn)C上在第一象限內(nèi)的點(diǎn).記l為經(jīng)過(guò)原點(diǎn)與點(diǎn)P的直線(xiàn),s為△DEM截直線(xiàn)l所得線(xiàn)段的長(zhǎng).試將s表示為直線(xiàn)l的斜率k的函數(shù).

Answer 1

解:(1)所求漸近線(xiàn)方程為y-=0,y+=0.                                 

(2)設(shè)P的坐標(biāo)為(x₀,y₀),則Q的坐標(biāo)為(-x₀,-y₀).                                   

λ=·=(x₀,y₀-1)·(-x₀,-y₀-1)

=-x₀²-y₀²+1=-x₀²+2.                                                        

∵|x₀|≥,∴λ的取值范圍是(-∞,-1］.                                          

(3)若P為雙曲線(xiàn)C上第一象限內(nèi)的點(diǎn),

則直線(xiàn)l的斜率k∈(0,).                                                 

由計(jì)算可得,當(dāng)k∈(0,］時(shí),s(k)=;

當(dāng)k∈(,)時(shí),s(k)=.                                       

∴s表示為直線(xiàn)l的斜率k的函數(shù)是

s(k)=