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

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 Parabola
Standard EquationDirectrixFocusLength of Latus rectumVertex
x = – aS: (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

Standard Parabola
CircleCenter is .

Radius is r.

Ellipse with horizontal major axisCenter 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 axisCenter 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 axisCenter Distance between the vertices is 2-a. Distance between the foci is 2-c .
Hyperbola with vertical transverse axisCenter 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)

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