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Given-x-4-16-y-2-2y-2-0-Show-that-is-a-reunion-of-and-Ellipsis-and-an-hyperbole-then-give-their-equations-

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Question Number 138470 by mathocean1 last updated on 13/Apr/21
Given (Γ): x^4 −16(y^2 −2y)^2 =0 Show that (Γ) is a reunion of and Ellipsis and an hyperbole then give their equations.
Given(Γ):x4−16(y2−2y)2=0Showthat(Γ)isareunionofandEllipsisandanhyperbolethengivetheirequations.
Answered by MJS_new last updated on 14/Apr/21
x^4 −16(y^2 −2y)^2 =0 (x^2 −4(y^2 −2y))(x^2 +4(y^2 −2y))=0 ⇒ (x^2 −4(y^2 −2y))=0 ∨ (x^2 +4(y^2 −2y))=0 x^2 −4(y^2 −2y)=0 ⇔ y=1±((√(x^2 +4))/2) hyperbola x^2 +4(y^2 −2y)=0 ⇔ y=1±((√(4−x^2 ))/2) ellipse
x4−16(y2−2y)2=0(x2−4(y2−2y))(x2+4(y2−2y))=0⇒(x2−4(y2−2y))=0∨(x2+4(y2−2y))=0x2−4(y2−2y)=0⇔y=1±x2+42hyperbolax2+4(y2−2y)=0⇔y=1±4−x22ellipse

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