In previous lecture we already mentioned that 'How developed the Conic Section is'?
There are several way to approach the study of conic. But now we are trying to introduce basically what conic section is? What are its types?Conic sections are commonly studied topic of geometry. They play an important role in both mathematics & physical process in nature.
Fig1: Conic Section Conic sections are the geometric curve formed when a cone is cutting by a plane. When double-napped cone is slicing by a plane, then different types of conic sections are obtained. A doubled-napped cone is 3-dimensional but formations of conic sections are 2-dimensional curves or plane curves. So the desirable definition of conic section avoid the notion of a cone.
The conic sections are divided into 4-major categories as follows:
1) Circle;
2) Ellipse;
3) Parabola;
4) Hyperbola.
1. Circle:
When a cone is cutting by the plane perpendicular to its conic axis, circle is formed.2. Ellipse:
When 3-dimensional cone is cutting by a plane which is not perpendicular to its conic axis or not parallel to the generator, but cutting a cone in a closed curve, ellipse is formed.
3. Parabola:
If 3-dimensional cone is cutting by a plane is parallel to one & only one generator then the conic is called a parabola.
4. Hyperbola:
If 3-dimensional cone is cutting by a plane is parallel with two generator of a cone, which intersects both napped of a cone. This types of conic is called a hyperbola.
 |
| Fig2: Conic Section |
Note that: The intersecting plane does not passing through the vertex of the cone. When plane does passes through the vertex the degenerating conic are formed. Degenerating conics: which include a point, a line, pair of parallel line & pair of intersecting lines. The figures are shown as follows:
 |
| Fig3: Degenerated Conic |
Definition of conic section:
A conic section is the locus of a point, which moves in a plane such that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line.
Fig4: Conic Geometrically
Where,
- The collection of points is called locus.
- The fixed point is called focus.
- The fixed line is called the directrix.
- The constant ratio is called the eccentricity. It is denoted by 'e'.
In the figure, '$P$' is a point & '$Q$' is the foot of a line from '$P$' perpendicular to the directrix. Where,
$\frac{FP}{PQ} = Constant = e \Rightarrow FP = ePQ$
The term eccentricity is defined as the ratio of focal distance to directrix distance of the conic section. It is a measurement of how much a conic deviated from being conic circular.
$eccentricity\;(e) = \frac{distance\;from\;focus\; to\;conic}{distance\;from\;conic\;to\;directrix}$
The conic sections are classified by its eccentricity as follows:
- If $e = 0$; the conic is a circle.
- If $e = 1$; the conic is a parabola.
- If $e < 1$; the conic is an ellipse.
- If $e > 1$; the conic is a hyperbola.
General equation of conic section:
Let us move from geometric conic to algebraic conic:The equation of the conic is represented by the general equation of 2nd degree.
Where $A$, $B$, $C$, $D$, $E$, $F$ are constant. If we change the value of these constants, the shape of the corresponding conic will also change. So we have to focus on these differences in the algebraic equations.The quantity $B^2 - 4AC = 0$ is the discriminant of above 2nd degree equation . If constant $A$, $B$, $C$ are not equal to zero, it can help to determine the shape of the conic.A conic sections can be classified by its discriminant as follows:
- If $C = A$ and $B = 0$, the conic is circle.
- If $B^2 - 4AC = 0$, the equation represents a parabola, a line or else has no graph.
- If $B^2 - 4AC < 0$, the equation represents an ellipse, a point or else has no graph.
- If $B^2 - 4AC > 0$, the equation represents a hyperbola, a pair of intersecting lines or else has no graph.
We will be discuss about 'Conic Sections & its Importance' later on.