# 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.

## 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 lineAxis 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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