Introduction to Conic Sections

Learning objectives:

This article focuses primarily on prerequisites to understand the conic sections and the real-time applications of conic sections.

The following is the brief of supporting concepts and the  features of this article:

  • Locus & Definition of Locus of points
  • Double  Cone.
  • Types of the conic sections
  • Definition of the eccentricity.
  • Focus & Directrix
  • The standard form of conic sections

Locus:-

Let us try to understand the concept of the locus with a related real-life example.

While driving a car, we obey traffic rules such as following the traffic signals, maintaining speed limits, slowing down near the school zones etc.

So all the moving vehicles on the road are guided by traffic rules & regulations.

Definition:

A Locus (basically a curve) is a set of points with coordinates (x,y) in a plane satisfying a given condition or rule. The family of points have well-defined criteria or property.

An equation of Locus:

The equation to the locus is the condition which the coordinates of each point of the locus are bound to satisfy.

Let us consider a few points on the graph as shown below.

 

These are the shattered or sprinkled points on the graph and these points do not follow a specific Rule or Pattern.

There is no rule or condition or an equation, which can guide these points in a meaningful shape or curve.

 

 

 

Contrary to earlier pictures, let consider below graphs where the points follow certain rules and pattern.

Set of points equidistant from a reference point.

 

It is evident from this graph that the red points follow a particular rule or a pattern.

We see that the set of all red points are at equidistance from the single reference black point and this forms a circular curve

So we say that the locus of points equidistant from a single reference point is a circle.

And these points follow the equation of the locus x^{2}+y^{2}=r^{2} where “r” is the radius of the circle.

 

Set of points equidistant from the two reference points

 

In this graph, we observe that the set of all red points are at an equidistant from the two reference black points.

So we can say that the locus of points that are equidistant from two reference points is a perpendicular bisector line joining the reference points.

The equation of the perpendicular bisector line can be found once the coordinates of the reference points are known.

 

Double Cone:

Definition of Cone:

Cone is a 3-dimensional picture with a circular base and tapers to a single point which is called  Apex. There are two kinds of cones

i) Oblique Circular Cone

ii) Right Circular Cone

 

The double Cone:

A Double cone consists of  2 right circular cones with their apex connected. There are two sections of the double cone as we see.

They are the upper nappe and lower nappe. The two nappes share the common axis of symmetry.

There is an angle  \alpha  formed between the axis of symmetry and the slant edge of the cone.

 This angle alpha is the reference angle for the formation of different conics from the double nappe.

When a plane intersects the axis of symmetry at an angle say \beta , the value of \beta taken will decide the type of conic section.

 

   Types of Conic Sections:

Let us say that a rectangular plane cuts the given conical nappe. The type of conic section formed depends on the angle at which the plane cuts the axis of symmetry.

Let us consider different angles at which the rectangular plane cuts the conical nappe.

 

      Circle formation in a Double Cone:

Circle formation in Double cone

 

 

In the picture to the left, we observe that a rectangular plane is in a horizontal position and is cutting the upper nappe of the double cone.

\beta is the angle formed between the rectangular plane and the axis of symmetry of the double nappe.

The rectangular plane is perpendicular to the axis of the symmetry which means, that the angle \beta =90^{\circ } . A circle is formed in the region where the rectangular plane cuts the upper nappe of the double cone.

\beta =90^{\circ } conveys that the conic formed would be a circle irrespective of the value of the angle \alpha.

 

 

Parabola formation in Double cone

 Parabola formation in a Double Cone:

 

Unlike the earlier picture, we observe that a rectangular plane is in an inclined position cutting the upper nappe of the double cone.

For a parabola to be formed from a double cone, there must be a relation between the angles \beta and \alpha .

When the angle made by rectangular plane \alpha =\beta the shape formed would be the parabola.

 

 

Ellipse formation in a double Cone:

Ellipse formation in double Cone

 

In this picture, we observe that a rectangular plane is in an inclined position cutting the upper nappe of the double cone similar to the parabolic conic section.

The only difference between the parabola formation and the ellipse formation is that the angle \beta   lies in between the angle \alpha and 90^{\circ } .

The conic section thus formed satisfying this criterion \alpha <\beta <90^{0}is the ellipse.

 

 

 

Hyperbola formation in Double cone:

Unlike earlier pictures, the rectangular plane intersects the upper and the lower nappe as well.

The magnitude of the angle alpha must be within 0 and the angle \alpha.The criteria for the formation of the hyperbola is  0<\beta <\alpha.

 

 

 

 

Eccentricity:-

In the previous section, we saw the formation of different conics when a rectangular plane cuts the double cone. The conic section formed changes as the angle \beta at which the rectangular plane cuts the double cone changes.

Yet there is another parameter called “eccentricity” that precisely defines the type of conic section.

The eccentricity of a conic section is a  number, which is always  \geq 0 . It is the measure of the deviation from the circular path.

Eccentricity “e” is a non-negative number ( e\geq 0) that uniquely characterises the conic sections.

Below table gives the value of the eccentricity of the 4 conic sections.

Eccentricity "e" of conic sections

Type of ConicValue of the Eccentricity "e"
Circle0
Parabola1
EllipseBetween 0 and 1
Hyperbolae>1
Straight lineInfinite

The Minimum value of eccentricity is “e” is for the circle and its value is 0.

More the eccentricity, more the deviation from the circular path. The eccentricity of an ellipse would be in between 0 and 1.

The parabola would have the eccentricity as 1 exactly and lastly, the hyperbola has the eccentricity “e”>1.

 

Focus & Directrix:-

Every conic section discussed above has an axis of symmetry. An Axis of symmetry divides the conic section into two equal halves. A focus is a point which lies on the axis of symmetry of a conic section.

A directrix is a straight line which is located outside the conic section and is perpendicular to the axis of symmetry of a conic section.

Neither the focus nor the directrix intersects the conic curve. They both stay away from the conic section.

But a point on the conic curve shares a relation with the focus and directrix of a conic.

At this point, we will be defining the value of the eccentricity “e”.The value of the eccentricity “e” is the ratio of the perpendicular distance from a point “p” on the conic section to the Directrix line and the distance between the focus to the point P.

The  picture below explains the various value of the  eccentricity for the  different  conic section

Conic sections and the eccentricity

 

Let us say that there is a common focal point “F” for the conic sections Ellipse, Parabola and Hyperbola.

And the points on the conic sections be P and the perpendicular distance from the fixed-line directrix be “M”.

The picture formed by maintaining the  “eccentricity e” as 1 is the parabola. \dfrac {FP}{PM}=1

Similar way, the picture formed such that “eccentricity e” is less than 1 is the  Ellipse.\dfrac {FP}{PM} <1

The picture which maintains the “eccentricity” greater than 1  is the hyperbola.\dfrac {FP}{PM} >1

 

The Standard Form of the conic section:-

There is a standard equation to represent a conic section. The equation is a second-degree equation which has two variables X and Y.

Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0.The type of the conic section is decided by the constants A,B,C,D,E,F

The value  B^{2}-4AC helps in identifying the type of conic section.

The table below gives the conditions for a particular conic to exist.

ConditionType of conic section
B^2-4AC <0Circle Or Ellipse
B^2 -4AC =0Parabola
B^2 -4AC >0Hyperbola

 

 

 

 

 

 

 

 

 

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