Answer
The six types of quadratic surfaces are elliptic paraboloid, hyperbolic paraboloid, cone, ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets.
Elliptic paraboloid: $\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$
Hyperbolic paraboloid: $\frac{z}{c}=\pm\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\pm\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\pm\frac{y^2}{b^2}+\frac{z^2}{c^2}$
Cone: $\frac{z^2}{c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y^2}{b^2}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x^2}{a^2}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$
Ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
Hyperboloid of one sheet: $-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
Hyperboloid of two sheets: $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ or, $-\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
Work Step by Step
The six types of quadratic surfaces are elliptic paraboloid, hyperbolic paraboloid, cone, ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets.
Elliptic paraboloid: $\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$
Hyperbolic paraboloid: $\frac{z}{c}=\pm\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\pm\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\pm\frac{y^2}{b^2}+\frac{z^2}{c^2}$
Cone: $\frac{z^2}{c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y^2}{b^2}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x^2}{a^2}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$
Ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
Hyperboloid of one sheet: $-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
Hyperboloid of two sheets: $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ or, $-\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
