Chapter 13 - Conic Sections - 13.3 Conic Sections: Hyperbolas - 13.3 Exercise Set - Page 871: 42

Ellipse

$4{{y}^{2}}+20{{x}^{2}}+1=8y-5{{x}^{2}}$ Combine like terms $4{{y}^{2}}+20{{x}^{2}}+5{{x}^{2}}-8y+1=0$ $25{{x}^{2}}+4{{y}^{2}}-8y+1=0$ Compare the equation $25{{x}^{2}}+4{{y}^{2}}-8y+1=0$ with the standard equation of a conic section $A{{x}^{2}}+B{{y}^{2}}+Cxy+Dx+Ey+F=0$, Here, $A=25,B=4$ Here, we see A and B have the same sign, but $A \ne B$. Therefore, the graph of the equation is an ellipse.