Chapter 10 - Analytic Geometry - 10.5 Rotation of Axes; General Form of a Conic - 10.5 Assess Your Understanding - Page 679: 43

hyperbola

The general equation of a conic in the form of $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ (i) defines a parabola if $B^2-4AC=0$ (ii) defines an ellipse if $B^2-4AC\lt0$ and $A\ne C$ (iii) defines a circle if $B^2-4AC\lt0$ and $A= C$ (iv) defines a hyperbola if $B^2-4AC\gt0$ Here $A=1,B=3,C=-2$, hence $B^2-4AC=3^4-4(1)(-2)=9+8=17\gt0$, thus it is a hyperbola.