**Conic sections** are one of the important topics in Geometry. It is a curve, generated by intersecting a right circular cone with a plane. It is termed as **‘conic’**. It has distinguished properties in Euclidean geometry.Â 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 Î², different kind of conic sections are obtained. Namely;

- Circle
- Ellipse
- Parabola
- Hyperbola

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, formulas and their significance.

## Conic Section Formulas

Check the formulas for different types of sections of cone in the table given here.

Circle | (xâˆ’a)^{2}+(yâˆ’b)^{2}=r^{2} |
Center isÂ (a,b)
Radius isÂ r |

Ellipse with horizontal major axis | (xâˆ’a)^{2}/h^{2}+(yâˆ’b)^{2}/k^{2}=1 |
Center isÂ (a, b) Length of the major axis isÂ 2h. Length of the minor axis isÂ 2k. Distance between the center and either focus isÂ cÂ with c ^{2}=h^{2}âˆ’k^{2}, h>k>0 |

Ellipse with vertical major axis | (xâˆ’a)^{2}/k^{2}+(yâˆ’b)^{2}/h^{2}=1 |
Center isÂ (a, b) Length of the major axis isÂ 2h. Length of the minor axis isÂ 2k. Distance between the center and either focus isÂ cÂ with c ^{2}=h^{2}âˆ’k^{2}, h>k>0 |

Hyperbola with horizontal transverse axis | (xâˆ’a)^{2}/h^{2}âˆ’(yâˆ’b)^{2}/k^{2}=1 |
Center isÂ (a,b) Distance between the vertices isÂ 2h Distance between the foci isÂ 2k. c ^{2}=h^{2}Â + k^{2} |

Hyperbola with vertical transverse axis | (xâˆ’a)^{2}/k^{2}âˆ’(yâˆ’b)^{2}/h^{2}=1 |
Center isÂ (a,b) Distance between the vertices isÂ 2h Distance between the foci isÂ 2kÂ . c ^{2}= h^{2}Â + k^{2} |

Parabola with horizontal axis | (yâˆ’b)^{2}=4p(xâˆ’a),Â pâ‰ 0 |
Vertex isÂ (a,b) Focus isÂ (a+p,b) Directrix is the line x=aâˆ’p Axis is the lineÂ y=b |

Parabola with vertical axis | (xâˆ’a)^{2}=4p(yâˆ’b),Â pâ‰ 0 |
Vertex isÂ (a,b) Focus isÂ (a+p,b) Directrix is the line x=bâˆ’p Axis is the line x=a |

### Focus, Eccentricity and Directrix of Conic

A conic section can also be described as the locus of a point P moving in the plane of a fixed point F known as **focus (F)** and a fixed line d known as **directrix** (with focus not on d) in such a way that the ratio of the distance of point P from focus F to its distance from d is a constant e known as **eccentricity**. Now,

- If eccentricity, e = 0, the conic is a circle,
- If 0<e<1, the conic is an ellipse
- If e=1, the conic is a parabola
- And if e>1, it is a hyperbola

So, eccentricity is a measure of the deviation of the ellipse from being circular. Suppose,Â the angle formed between the surface of the cone and its axis isÂ Î² and the angle formed between the cutting plane and the axis is Î±, the eccentricity is;

**e = cos Î±/cosÂ Î²**

### Parameters of Conic

Apart from focus, eccentricity and directrix, there are few more parameters defined under conic sections.

**Principal Axis:**Line joining the two focal points or foci of ellipse or hyperbola. Its midpoint is the centre of the curve.**Linear Eccentricity:**Distance between the focus and centre of a section.**Latus Rectum:**A chord of section parallel to directrix, which passes through a focus.**Focal Parameter:**Distance from focus to the corresponding directrix.**Major axis:**Chord joining the two vertices. It is the longest chord of an ellipse.**Minor axis:**Shortest chord of an ellipse.

**Also, read:**

### Sections of the Cone

Consider a fixed vertical line â€˜lâ€™ and another line â€˜mâ€™ inclined at an angle â€˜Î±â€™ intersecting ‘l’ at point V as shown below:

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

- V is the vertex of the cone
- l is the axis of the cone
- m, the rotating line the is a generator of the cone

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

### Conic Section Circle

If Î²=90^{o}, the conic section formed is a circle as shown below.

### Conic Section Ellipse

If Î±<Î²<90^{o}, the conic section so formed is an ellipse as shown in the figure below.

### Conic Section Parabola

If Î±=Î², the conic section formed is a parabola (represented by the orange curve) as shown below.

**Conic Section Hyperbola**

If 0â‰¤Î²<Î±, then the plane intersects both nappes and conic section so formed is known as a hyperbola (represented by the orange curves).

### Conic Section Standard Forms

After the introduction of Cartesian coordinates, the focus-directrix property can be utilised to write the equations provided by the points of the conic section. When the coordinates are changed along with the rotation and translation of axes, we can put these equations into standard forms. For ellipses and hyperbolas, the standard form has the x-axis as the principal axis and the origin (0,0) as the centre. The vertices are (Â±a, 0) and the foci (Â±c, 0). Define b by the equations c^{2}= a^{2}Â âˆ’ b^{2}Â for an ellipse and c^{2} = a^{2} + b^{2} for a hyperbola.

For a circle, c = 0 so a^{2} = b^{2}. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix is the line with equation x = âˆ’a. In standard form the parabola will always pass through the origin.

- Circle: x
^{2}+y^{2}=a^{2} - Ellipse: x
^{2}/a^{2}+ y^{2}/b^{2}= 1 - Hyperbola: x
^{2}/a^{2}– y^{2}/b^{2}= 1 - Parabola: y
^{2}=4ax when a>0

### Conic Sections Examples

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

- If Î±< Î²â‰¤90
^{Â°}, then the plane intersects the vertex exactly at a point.

- If Î±=Î², 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.

- If 0â‰¤Î²<Î±, the section formed is a pair of intersecting straight lines. This condition is a degenerated form of a hyperbola.

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