In this section we will discuss some of the basic geometric properties of parabolas, ellipses, and hyperbolas. These curves play an important role in calculus and also arise naturally in a broad range of applications in such fields as planetary motion, design of telescopes and antennas, global positioning and medicine to name a few.
Conic sections
Circles, ellipses parabolas and hyperbolas are called conic sections or conics because they can be obtained as intersections of a plane with a double-napped circular cone. (See figure below.)

Definitions of the conic sections
Parabola
A parabola is the set of all points in the plane that are equidistant from a fixed line and a fixed point not on the line.
The line is called the directrix of the parabola, and the point is called the focus. ( See figure below). A parabola is symmetric about the line that passes through the focus at right angle to the directrix. This line is called the axis or the axis of symmetry of the parabola, intersects the parabola at a point called the vertex.
Ellipse
An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points is a given positive constant that is greater than the distance between the fixed points . The two fixed points are called the foci of the ellipse, and the midpoint of the line segment joining the foci is called cente r. (see picture below)

Hyperbola
A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed distinct points is a given positive constant that is less than the distance between the fixed points.
The two fixed points are called the foc i of the hyperbola, and the term “difference” that is used in the definition is understood to mean the distance to the farther focus minus the distance to the loser focus . As a result the points on the hyperbola form two branches , each “wrapping around “the closer focus. (See figure below). The midpoint of the line segment joining the foci is called center of the hyperbola, the line through the foci is called the focal axis , and the line through the center that is perpendicular to the focal axis is called the conjugate axis . The hyperbola intersects the focal axis at two points called the vertices. Associated with every hyperbola ia a pair of lines, called the asymptotes of the hyperbola. These lines intersect at the center of the hyperbola and have the property that as a point P moves along the hyperbola away from the center, the vertical distance between P and one of the asymptotes approaches zero.

Equations of Parabolas in Standard Position
It is traditional in the study of parabolas to denote the distance between the focus and the vertex by p. The vertex is equidistant from the focus and the directrix, so the distance between the vertex and the directrix is also p; consequently, the distance between the focus and the directrix is 2p (see figure below). As illustrated in that figure, the parabola passes through two of the corners of a box that extends from the vertex to the focus along the axis of symmetry and extends 2p units above and 2p units below the axis of symmetry. The equation of a parabola simplest if the vertex is the origin and the axis of symmetry is along the x-axis or y-axis. The four possible such orientations are shown in the figure below. These are called the standard positions of a parabola, and the resulting equations are called the standard equations of a parabola.
To derive the equation for a parabola with focus (p , 0) and directrix x = -p. Let P(x , y) be a point on the parabola. Since P is equidistant from the focus and directrix, the distances PF and PD are equal; That is,
PF = PD
You can use this equation to derive the equation of the parabola lying on the x-axis.

Equations of ellipses in standard position
It is traditional in the study of ellipses to denote the length of the major axis by 2a, and the length of the minor axis by 2b, and the distance between the foci by 2c (see figure) .
There is a classic relationship between the numbers a, b, and c that can be obtained by examining the sum of the distances to the foci from a point at the end of the major axis and from point Q at the end of the minor axis. These sums must be equal so we obtain
2 b
2  c
2 
( a
 c )

( a
 c )
From which it follows that a
 b
2  c
2 or c
 a
2  b
2
From these relations it can be shown that the distance from a focus to an end of the minor axis is a , which implies that for all points on the ellipse the sum of the distances to the foci is 2a. It also follows from the relations that a
 b with the equality holding only when c = 0. Geometrically, this means that the major axis of an ellipse is at least as large as the minor axis and that the two axes have a equal length only when the foci coincide, in which cases the ellipse is a circle.
The equation of an ellipse is simplest if the center of the ellipse is at the origin and the foci are on the x-axis or the y-axis. The two possible orientations are shown below. These are called the standard positions of an ellipse, and the resulting equations are called the standard equations of an ellipse.
You can derive the equation for an ellipse with foci on the x-axis using the relation:
P F
 
PF

2 a and the fact that a
 b
2  c
2
.

Equations of hyperbolas in standard position
It is traditional in the study of hyperbolas to denote the distance between the vertices by
2a, and the distance between the foci by 2c, and to define b as b
 c
2  a or equivalently c
 a
2  b
2
(see picture)
As illustrated in the picture the asymptotes pass through the corners of a box extending b units on each side of the center along the conjugate axis and units on each side of the center along the focal axis. The number a is called the semi focal axis of the hyperbola and the number b is called the semi conjugate axis.
The equation of a hyperbola can be derived from the following relations
P F
 
PF

2 a And c
 a
2  b
2
The equations of the asymptotes are y
  b a x