Chapter 10 - Analytic Geometry - 10.5 Rotation of Axes; General Form of a Conic - 10.5 Assess Your Understanding - Page 679: 47

parabola

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 $A=9,B=12,C=4$, hence $B^2-4AC=(12)^2-4(9)(2)=144-144=0$, thus it is a parabola.