Conic sections are the curves formed when a plane intersects the surface of a right circular cone. Except in some degenerate cases, the curves formed are the circle, the ellipse, the parabola, and the hyperbola.

1. Definition. A parabola is the set of all points in the plane that are the same distance from a fixed line and a fixed point not on the line. The fixed line is called the directrix, and the fixed point is called the focus.

2. Axis or axis of symmetry. This is the line that passes through the focus and is perpendicular to the directrix.

3. Vertex. This is the point at which the parabola intersects its axis. The vertex is halfway between the focus and the directrix.

4. Standard forms of equations of parabolas with vertex at (0, 0) and $\mathit{p}\mathbf{>}\mathbf{0}$ are

   $y^2=\pm 4px$ and $x^2=\pm 4py.$ (See the table on page 663)

5. The latus rectum of a parabola is the line segment that passes through the focus, is perpendicular to the axis of the parabola, and has endpoints on the parabola. The length of the latus rectum (focal diameter) of a parabola $y^2=\pm 4px$ or $x^2=\pm 4py$ is 4p.

6. Standard forms of equations of parabolas with vertex (h, k) and $\mathit{p}\mathbf{>}\mathbf{0}$ are ${(y-k)}^2=\pm 4p(x-h)$ and ${(x-h)}^2=\pm 4p(y-k).$ (See the table on page 667.)

1. Definition. An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points is a constant. The fixed points are called the foci of the ellipse.

2. Vertices. These are the two points where the line through the foci intersects the ellipse.

3. Major axis. This is the line segment joining the two vertices of the ellipse.

4. Center. This is the midpoint of the line segment joining the foci. It is also the midpoint of the major axis.

5. Minor axis. This is the line segment that passes through the center of the ellipse, is perpendicular to the major axis, and has endpoints on the ellipse.

6. Standard forms of equations of ellipses with center $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathit{a}\mathbf{>}\mathit{b}\mathbf{>}\mathbf{0}\mathbf{,}\mathit{c}\mathbf{<}\mathit{a}\mathbf{,}$ and ${\mathit{b}}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{-}{\mathit{c}}^{\mathbf{2}}$ are

   $${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1,\text{ with foci }(\pm c,0)\text{ and vertices }(\pm a,0),$$

   and

   $${\displaystyle \frac{x^2}{b^2}}+{\displaystyle \frac{y^2}{a^2}}=1,\text{ with foci }(0,\pm c)\text{ and vertices }(0,\pm a).$$

   See the table on page 678.

7. Standard forms of equations of ellipses with center (h, k), $\mathit{a}\mathbf{>}\mathit{b}\mathbf{>}\mathbf{0}\mathbf{,}$ and ${\mathit{b}}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{-}{\mathit{c}}^{\mathbf{2}}$ are

   ${\displaystyle \frac{{(x-h)}^2}{a^2}}+{\displaystyle \frac{{(y-k)}^2}{b^2}}=1$ and ${\displaystyle \frac{{(x-h)}^2}{b^2}}+{\displaystyle \frac{{(y-k)}^2}{a^2}}=1.$

   See the table on page 680.

1. Definition. A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points is a constant. The fixed points are called the foci of the hyperbola.

2. Vertices. These are the two points where the line through the foci intersects the hyperbola.

3. Transverse axis. This is the line segment joining the two vertices of the hyperbola.

4. Center. This is the midpoint of the line segment joining the foci. It is also the midpoint of the transverse axis.

5. Standard forms of equations of hyperbolas with center (0, 0), $\mathit{c}\mathbf{>}\mathit{a}\mathbf{,}$ and ${\mathit{b}}^{\mathbf{2}}\mathbf{=}{\mathit{c}}^{\mathbf{2}}-{\mathit{a}}^{\mathbf{2}}$ are

   $${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1,\text{ with foci }(\pm c,0)\text{ and vertices }(\pm a,0)$$

   and

   $${\displaystyle \frac{y^2}{a^2}}-{\displaystyle \frac{x^2}{b^2}}=1,\text{ with foci }(0,\pm c)\text{ and vertices }(0,\pm a).$$

   See the table on page 692.

6. Conjugate axis. This is a line segment of length 2b that passes through the center of the hyperbola and is perpendicularly bisected by the transverse axis.

7. Asymptotes. The asymptotes of the hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$ are the two lines $y=\pm {\displaystyle \frac{b}{a}}x$; the asymptotes of the hyperbola ${\displaystyle \frac{y^2}{a^2}}-{\displaystyle \frac{x^2}{b^2}}=1$ are the lines $y=\pm {\displaystyle \frac{a}{b}}x.$

8. Graphing. A procedure for graphing a hyperbola centered at (0, 0) is given on page 697.

9. Standard forms of equations of hyperbolas centered at (h, k) are

   $${\displaystyle \frac{{(x-h)}^2}{a^2}}-{\displaystyle \frac{{(y-k)}^2}{b^2}}=1\text{ and }{\displaystyle \frac{{(y-k)}^2}{a^2}}-{\displaystyle \frac{{(x-h)}^2}{b^2}}=1.$$

   See the table on page 698. Also see page 699 for a procedure for graphing such hyperbolas.