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State And Explain The Equation That Drives An Ellipse

Referencing Styles : Harvard | Pages : 1

Ellipse in the field of Mathematics, can be described as a Geometrical figure that is basically a plain curve and typically surrounds two focal points. The curve surround the two focal points in a manner that for all the other points on the curve, the sum of the distances in between the two focal points are uniform and do not vary. In other words, the geometrical figure is similar to a circle which is a special type of ellipse. Typically, in a circle the two of the focal points are similar. It should further be noted in this context that the elongation of an ellipse is generally measured by the eccentricity, which is denoted by the number e. the value of e ranges from zero to one. e = 0 represents the limiting case of a circle whereas e = 1 signifies infinite elongation in which an ellipse is no longer possible but a parabola can be possible.

In analytical manner, the standard equation of the ellipse is generally centred on the width and origin of the ellipse and the equation is presented below:

 

The standard parametric equation for ellipse is:

(x, y) = (a cos (t), b sin (t)) for 0 ≤ t ≤ 2π

Generally, ellipses are considered as closed type conic section. At times, ellipses might have similarities with the other type of conic sections which are hyperbolas and parabolas. On the other hand, an ellipse can also be defined on the term of focus point and directrix (line outside of an ellipse).

It should further be noted that for all the points located within an ellipse, the ration between the distance to the directrix and the distance to the focus remain constant. The constant ration within the above mentioned eccentricity can be found by the following equation,

 

Ellipses are extremely common in the subjects of Astronomy, Engineering and Physics. For instance, the specific orbital plane of each planet around which the planets revolve within the solar system is the form of an ellipse. The sun is placed at one focal point which is also known as the focus or the barycentre of the sun and planet pair. It should be noted further that the similar view point holds true for the moon and all other orbiting planets. Astronomy states that the shape of the planets and the stars are compared equivalent to that of ellipsoids. It should be noted in this context that the shape of a circle when viewed from a certain angle appears equivalent to an ellipse. The ellipse is therefore an image of a circle under a perspective or parallel projection. In addition to this, it can also be mentioned that an ellipse is one of the simplest Lissajous figure which is formed due to the horizontal and vertical motions of the sinusoids with the constant frequency. The similar phenomenon contributes to elliptical polarization of light in the field of optics. The historical background studies state that the name ellipse was propounded by Apollonius of Perga in his account Conics.

An ellipse is a geometrical figure which is based on the locus of points within a Euclidean plane. Considering two fixed points, F1 and F2 which are known as the foci and the distance being equivalent to 2 a which is more than the distance between the two foci, and the ellipse is a result of all of the set of points or the sum of distances of the foci and is denoted by P.

 

The midpoint which is denoted as C of the line segment joins the foci at the centre of the ellipse. The line that passes through the foci is known as the major axis and the line that is perpendicular to it and passes through the centre is known as the minor axis. It should be noted further that the major axis typically intersects the ellipse at the two vertex points that are denoted as V1 and V2. The distance between the two points is denoted by a, from the centre. The distance C, of the foci to the centre is referred to as the focal distance or linear eccentricity and the quotient e is represented as e= c/a and is the formula to calculate eccentricity.

In case, if the F1 is equivalent to F2, a circle is formed and circle forms a special type of ellipse. The equation |PF2| + |PF1| = 2a and can be viewed as a circle. Considering other condition where C2 is a circle and has the midpoints F2 and 2 a, the distance of the point P to the circle C2 is equivalent to the focus F1 such that |PF1| = |Pc2|. C2 in this case is known as the circular directrix which is closely related to the focus F2 of an ellipse. However, it is integral not to confuse the definition of an ellipse that makes use of directrix line.

With the use of Dandelian spheres, it can be proven that any plane section of a cone with a plane is a perfect ellipse with the assumption that the plan does not comprise of an apex and has a gentle slope compared to the lines on the cone.

The standard form of an ellipse in the field of Cartesian coordinates assumes that the origin acts as the centre of the ellipse and that X- axis is the major axis and the foci points are the points,

 

And the vertices are V1 = (a,0) and V2 = (-a,0)

Considering an arbitrary point (x,y), the distance to the focus (C,0) can be given as   and to the other focus . Therefore the point (x,y) is on the ellipse whenever,

 

Removing the radicals by means of squaring and using the formula, b2= a2-c2, the standard equation for the ellipse can be deduced which is equivalent to,

 

 

The equation can be solver for Y with the help of equation listed below,

 

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