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Conic Sections Standard Form: Types: Terminology: Standard Parabola
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)