橢圓x^2+y^2/4=1/4及直線y=x+m
⑴當(dāng)直線和橢圓有共公點(diǎn),求實(shí)數(shù)m的取值范圍.
y=x+m代入橢圓方程中得:4x^2+x^2+2mx+m^2-1=0
5x^2+2mx+m^2-1=0
判別式=4m^2-20(m^2-1)>=0
16m^2<=20
m^2<=5/4
-根號(hào)5/2<=m<=根號(hào)5/2.
⑵求被橢圓截得的最長(zhǎng)玄所在的直線方程.
x1+x2=-2m/5,x1x2=(m^2-1)/5
弦長(zhǎng)=根號(hào)(1+k^2)*|x2-x1|=根號(hào)(1+1)*根號(hào)[(x1+x2)^2-4x1x2]=根號(hào)2*根號(hào)(4m^2/25-4(m^2-1)/5)=根號(hào)2*根號(hào)[(-16m^2+20)/25]
所以,當(dāng)m=0時(shí),弦長(zhǎng)為最長(zhǎng).此時(shí)直線方程是y=x.