# Conic Sections Standard Form: Types: Terminology: Standard Parabola (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Title: Conic Sections Standard Form

- A conic section is the locus of a point moving in a plane such that its distance from a fixed point (focus) is in a constant ratio to its perpendicular distance from a fixed line.
- In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane.
- The constant ratio is called eccentricity of the conic.
- A
**conic section**is the intersection of a plane and a double right circular cone. - By changing the angle and location of the intersection, we can produce different types of conics.
- There are four basic types: circles, ellipses, hyperbolas, and parabolas.
- None of the intersections will pass through the vertices of the cone.

## Types of Conic Sections

## Terminology

**Axis of conic**: Line passing through focus, perpendicular to the directrix.**Vertex**: Point of the intersection of conic and axis.**Chord**: Line segment joining any 2 points on the conic.**Double ordinate**: Chord perpendicular to the axis**Latus Rectum**: Double ordinate passing through focus.- The general equation for any conic section is
- Where A, B, C, D, E and F are constants.
- As we change the values of some of the constants, the shape of the corresponding conic will also change.
- It is important to know the differences in the equations to help quickly identify the type of conic that is represented by a given equation.
- If is less than zero, if a conic exists, it will be either a circle or an ellipse.
- If equals zero, if a conic exists, it will be a parabola.
- If is greater than zero, if a conic exists, it will be a hyperbola.

### Standard Parabola

Standard Equation | Directrix | Focus | Length of Latus rectum | Vertex |

x = – a | S: (a, 0) | 4-a | (0,0) | |

x = a | (- a, 0) | 4-a | (0,0) | |

y = – a | (0, + a) | 4-a | (0,0) | |

y = a | (0, -a) | 4-a | (0,0) |

### Standard Parabola Examples

Circle | Center is . Radius is r. | |

Ellipse with horizontal major axis | Center is . Length of major axis is 2-a. Length of minor axis is 2-b. Distance between center and either focus is cc with | |

Ellipse with vertical major axis | Center is Length of major axis is 2-a. Length of minor axis is 2-b. Distance between center and either focus is cc with | |

Hyperbola with horizontal transverse axis | Center Distance between the vertices is 2-a. Distance between the foci is 2-c . | |

Hyperbola with vertical transverse axis | Center is Distance between the vertices is 2-a. Distance between the foci is 2-c. | |

Parabola with horizontal axis | Vertex is Axis is the line | |

Parabola with vertical axis | Vertex is . Focus is Directrix is the line Axis is the line |

## Important Results of a Parabola

- 4 x distance between vertex and focus = Latus rectum = 4-a.
- 2 x Distance between directrix and focus = Latus rectum = 2 (2-a) .
- Point of intersection of Axis and directrix and the focus is bisected by the vertex.
- Focus is the midpoint of the Latus rectum.
- (Distance of any point on parabola from axis)
^{2}= (LR) (Distance of same point from tangent at vertex)