Chapter 13 - Conic Sections - 13.3 Conic Sections: Hyperbolas - 13.3 Exercise Set - Page 871: 43

Ellipse

$16{{x}^{2}}+5{{y}^{2}}-12{{x}^{2}}+8{{y}^{2}}-3x+4y=568$ Combine like terms $16{{x}^{2}}-12{{x}^{2}}+5{{y}^{2}}+8{{y}^{2}}-3x+4y=568$ $4{{x}^{2}}+13{{y}^{2}}-3x+4y-568=0$ Compare the equation $4{{x}^{2}}+13{{y}^{2}}-3x+4y-568=0$ with the standard equation of a conic section$A{{x}^{2}}+B{{y}^{2}}+Cxy+Dx+Ey+F=0$, Here, $A=4,B=13$ Here, A and B have the same sign, but $ A \ne B $. Therefore, the graph of the equation is an ellipse.