If cos α=23, then the range of values of ϕ on the ellipse
x2+4y2=4 falls inside the circle x2+y2+4x+3=0 is
π−α,π+α
The point ϕ on the ellipse x24+y2=1 will be (2 cos ϕ,sin ϕ).
It lies inside the circle x2+y2+4x+3=0 if
4 cos2ϕ+sin2ϕ+8 cosϕ+3<0⇒3 cos2ϕ+8 cosϕ+4<0⇒(cos ϕ+2)(3 cos ϕ+2)<0or −2<cos ϕ<−23or −1≤cos ϕ<cos(π−α) (∵ cos ϕ≮−1)∴π−α<ϕ<π+α is most appropriate.