Conic Sections - Coordinate Geometry

Conic sections are one of the important topics in Geometry. It is basically a curve, generated by intersecting a right circular cone with a plane. It forms four types of figure, a circle, an ellipse, parabola, and hyperbola. The details of these types will be learned more here in this article and also get the pdf of it at the end.

Conic Section Formulas

Circle (x−a)2+(y−b)2=r2(x−a)2+(y−b)2=r2 Center is (a,b)

Radius is r

Ellipse with horizontal major axis (x−a)2/h2+(y−b)2/k2=1 Center is (a, b)
Length of the major axis is 2h.
Length of the minor axis is 2k.
Distance between the center and either focus is c with
c2=h2−k2, h>k>0
Ellipse with vertical major axis (x−a)2/k2+(y−b)2/h2=1 Center is (a, b)
Length of the major axis is 2h.
Length of the minor axis is 2k.
Distance between the center and either focus is c with
c2=h2−k2, h>k>0
Hyperbola with horizontal transverse axis (x−a)2/h2−(y−b)2/k2=1 Center is (a,b)
Distance between the vertices is 2h
Distance between the foci is 2k.
c2=h2  + k2
Hyperbola with vertical transverse axis (x−a)2/k2−(y−b)2/h2=1 Center is (a,b)
Distance between the vertices is 2h
Distance between the foci is 2k .
c2= h2  + k2
Parabola with horizontal axis (y−b)2=4p(x−a), p≠0 Vertex is (a,b)
Focus is (a+p,b)
Directrix is the line
x=a−p
Axis is the line y=b
Parabola with vertical axis (x−a)2=4p(y−b), p≠0 Vertex is (a,b)
Focus is (a+p,b)
Directrix is the line
x=b−p
Axis is the line x=a

The rear mirrors you see in your car or the huge round silver ones you encounter at a metro station. These mirrors are examples of curves. Curves have huge applications everywhere, be it the study of planetary motion, the design of telescopes, satellites, reflectors etc. Conic consist of curves which are obtained upon the intersection of a plane with a double-napped right circular cone. It has been explained widely about conic sections in class 11. Let us discuss the formation of different sections of the cone and their significance.

The vertex of the cone divides it into two nappes referred to as the upper nappe and the lower nappe. In figure B, the cone is intersected by a plane and the section so obtained is known as a conic section. Depending upon the position of the plane which intersects the cone and the angle of intersection β, different kind of conic sections are obtained. Namely;

  • Circle
  • Ellipse
  • Parabola
  • Hyperbola

Let us briefly discuss the different conic sections formed when the plane cuts the nappes (excluding the vertex).

Conic Sections Circle

If β=90o, the conic section formed is a circle as shown below.

Conic Sections - Circle

Conic Section Ellipse

If α<β<90o, the conic section so formed is an ellipse as shown in the figure below.

Conic Sections -Ellipse

Conic Sections Parabola

If α=β, the conic section formed is a parabola (represented by the orange curve) as shown below.

Conic Section-ParabolaConic Section Hyperbola

If 0≤β<α, then the plane intersects both nappes and conic section so formed is known as a hyperbola (represented by the orange curves).

Conic Sections-Hyperbola

Conic Sections Examples

If the plane intersects exactly at the vertex of the cone, the following cases may arise:

  • If α< β≤90°, then the plane intersects the vertex exactly at a point.

Conic Section- Plane intersects at vertex

  • If α=β, the plane upon an intersection with cone forms a straight line containing a generator of the cone. This condition is a degenerated form of a parabola.

Conic Section-Cone generation

  • If 0≤β<α, the section formed is a pair of intersecting straight lines. This condition is a degenerated form of a hyperbola.

Conic Section-Pair of Intersecting straight lines

Sections of the Cone

Consider a fixed vertical line ‘l’ and another line ‘m’ inclined at an angle ‘α’ intersecting ‘l’ at point V as shown below:

conic section

conic sections explanation

The initials as mentioned in the above figure A carry the following meanings:

  1. V is the vertex of the cone
  2. l is the axis of the cone
  3. m, the rotating line the is a generator of the cone

 

Conic Section Equations

  • The equations of Ellipse is – x2/p2 + y2/q2 = 1
  • In case of the circle, p=q=radius, therefore – x2/p2 + y2/p2 = 1
  • The equation of Hyperbola is – x2/p2 – y2/q2 = 1

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