Several types of conic sections can be obtained depending on the point of intersection of the plane with the plane and the angle generated by the cone’s vertical axis. The term eccentricity is defined in terms of a fixed-point called focus and a fixed-line in the plane called the directrix. We will go through the eccentric meaning in geometry, as well as the eccentricity formula and the eccentricity of various conic sections such as parabola, ellipse, and hyperbola, with solved examples.
The form is distinguished by the eccentricity in the conic section, which should have a non-negative real value. Eccentricity, in general, is a measure of how much the curve has departed from the circularity of the given shape.
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What is a Hyperbola?
When a circular cone intersects with a plane that cuts the two nappes (see cone) of the cone produces a hyperbola, a two-branched open curve with a conic section. It may be described as the route (locus) of a point traveling in such a way that the ratio of the distance is measured from a fixed point (the focus) to the distance measured from a fixed line (the directrix) is a constant bigger than one as a plane curve. On the other hand, the hyperbola has two foci due to its symmetry.
A moving point with a consistent difference in distance from two fixed sites, or foci, is another definition. A degenerate hyperbola is formed when a circular cone intersects with a plane that cuts both nappes of the cone via the apex (two intersecting lines).
The hyperbola’s transverse axis is a line drawn through the foci and beyond; the conjugate axis is perpendicular to that axis and meets it at the geometric center of the hyperbola, a position midway between the two foci. The hyperbola is symmetrical along both axes.
If you want to know more about eccentricity as well as hyperbolas you can visit the Cuemath website.
Hyperbola’s Components
Let’s go through a few important words related to the different characteristics of a hyperbola.
Hyperbola foci: The hyperbola has 2 foci, with coordinates F(c, o) and F’ (-c, 0).
Center of Hyperbola: The center of the hyperbola is the middle of the line connecting the two foci.
Major Axis: The hyperbola’s major axis measures 2a units in length.
Minor Axis: The hyperbola’s minor axis measures 2b units in length.
Vertices: The vertices are the spots on the hyperbola where it crosses the axis. The hyperbola’s vertices are (a, 0), (-a, 0).
Hyperbola in Real Life
- The hyperbola form is often used in bridge design.
- Some comets’ open orbits around the Sun follow hyperbolas.
- The hyperbolic character of the interference pattern is formed by two circular waves.
Formula for Eccentricity
In an elliptical orbit, the planets circle around the Earth. Earth’s orbit has a lower eccentricity (e = 0.0167) than Mars’ (e=0.0935). The form appears less and less like a circle as the eccentricity value advances away from zero. A parabola has one focus and one directrix, whereas an ellipse and hyperbola have two foci and two directrixes, respectively. In the case of an ellipse, their eccentricity formulae are provided in terms of their semimajor axis(a) and semiminor axis(b), while in the case of a hyperbola, a equals semi-transverse axis and b equals semi-conjugate axis. The eccentricity equation is as follows:
Distance to the focus/Distance to the directrix equals eccentricity.
where, e = c/a
In the equation given above, e stands for eccentricity, c stands for the distance between any two points on the conic section and the focus, a stands for the distance between any two points on the conic section and their directrix.
