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Foci Vertices Axes And Center Of An Ellipse: Equations Of Ellipses Centered At The Origin

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Geometry is often considered to be one of the most complicated subjects of the world because of the large number of shapes with which the individuals are required to deal with. In this regard, mention needs to be made of the shapes of circle, triangle, square, rectangle, ellipse and others and the most complicated part is the fact that all these different figures have various kinds of formulae without which the equations related to them cannot be solved. It is true that some of the figures or the shapes are almost identical in nature and therefore some of the formulae of those two shapes are almost identical in nature however in the particular context of the majority of the shapes it is seen that the individuals have no other option but to memorize the different formulae related to them. In addition to these, it is seen that students are also required to effectively memorize the different theorems or the equations of trigonometry and other branches of knowledge for effective solving the diverse problems related to the above mentioned shapes.   

Ellipse is one of the most commonly used shapes in the subject of geometry and one of the major reasons for this can be ascribed to the fact that there are many in nature which are actually perfectly elliptical in nature. As a matter of fact, even the solar system along with the paths in which the different planets orbit around the sun is elliptical in nature. This as a matter of fact offers an insight into the reason why the shape under discussion here is one of the most commonly used ones in the geometry and also why extensive research had been undertaken on the concerned shape over the years. In this regard, it needs to be said that attempts have been made to solve the different problems related to the ellipses through the usage of the knowledge of physics since it is likely to help them to solve the equations in a much simpler manner.

The individuals for the purpose of effectively dealing with the shapes of ellipse need to have an effective knowledge regarding different concepts related to the same. Some of the most important concepts related to the ellipse in this regard are axes, foci, vertices, centre and others. In this regard, it needs to be said there are certain specific formulas for the analysis as well as for the for the purpose of finding the exact value of these attributes of the ellipse. However, before dealing or for that matter trying to understand the formulas for finding the values of these attributes of the ellipse the individuals are likely to gain in a substantial manner if they try to understand the meaning or the definition of these attributes or terms.

The centre of an ellipse can be defined as the exact point of origin of the ellipse which as a matter of fact is equidistant from the other points which are being placed on the circumference of the same. In this regard, it needs to be said that the centre of an ellipse is almost similar to the centre of the circles although there is a slight difference between the two shapes. As a matter of fact, it is seen that the centre of the ellipse is the fulcrum point of the entire shape and if a concrete physical object is being created which is elliptical in shape then it is likely that its centre would be the centre of mass of the entire physical object as well. More importantly, it had been seen that the coordinates at the centre of an ellipse just like that of the circle is usually considered to be zero and it is from this point onwards the calculation begins.

An important aspect to note regarding the ellipses is the fact that rather than containing just one focus, they consists of two foci which are located at an equal distance from the centre of the same. However, the problem arises because of the fact that the ellipses are of two types, namely horizontal ellipses and vertical ellipses and depending on the nature of the ellipse which is being taken into account the equation for the foci of the ellipse also changes. For example, the equations which are generally being used in the particular context of the horizontal and vertical ellipses for the calculation of the foci of the same are given below-

 

Furthermore, if the individuals are not comfortable with the usage of the above given complex equations then they even have the option to use the generic ellipse equation which over the years had been used for the purpose of calculating the foci of the ellipses. The equation that the individuals can use is given below-

c2 = a2 - b2

The axes of the ellipse, on the other hand, is generally defined as the major axis of the ellipse which connects the two straight lines on the two vertices of the ellipses. In this regard, it needs to be said that this is one of the most important features of the ellipses and the circles which distinguishes them from the other shapes commonly found in the subject of geometry or for that in the nature. As a matter of fact, it is seen that the length of the axis of the ellipse is usually considered to be half the length of the entire diameter of the concerned ellipse and thereby by having an idea regarding the length of the axis of the ellipse the individuals have the opportunity to not only calculate the diameter of the ellipse but at the same time locate the exact centre of the concerned ellipse as well.

The formula which is usually used for calculating the axis of the ellipse or rather the length of the ellipses is given below-

c2=a2−b2

The above equation in a succinct manner indicates the fact it is almost similar to the equation which is being used for calculating the foci of the ellipses however the only difference is the fact that in the place of the addition symbol there is a subtraction symbol.

 The different similarities between the circles and the ellipses not withstanding there is one most important difference between the two. For example, it is seen that the circle can have only one point of origin whereas the ellipses can have multiple points of origin. In this regard, it needs to be said that it is not at all necessary for the ellipses to have their point of origin at the centre of the ellipse. As a matter of fact, the ellipses can have their point of origin not only at its centres but also at the other points within the ellipse itself. More importantly, over the years it had been seen that mathematicians have come forward with different kinds of theorems and equations which in turn try to explain the nature of the ellipses which do not have their point of origin at the centre of the same.   

The equation of ellipse which is commonly being used for the ellipses that are centred at the origin of the ellipse is being outline below-

 

Furthermore, the individuals even have the option to take the help of the below given equation for the analysis of the ellipses which have their centers as their point of origin-

 

In contrast to these, it had been seen that there are various other kinds of ellipses which do not have their centers as their points of origin. In this regard, it needs to be said that these kinds of ellipses are rather uncommon in nature and the students are required to work with only in the latter part of their education or rather at the research level. This can be explained on the basis of the fact that they are rather complex in nature and thus the individuals often find it difficult to solve problems related to them or for that matter even handle them. The equation which is being generally used for the ellipses which do not have their centers as their point of origin is highlighted below-

 

In this regard, it needs to be said that one of the most important difficulty that the individuals face while solving this equation especially designed or for that matter formulated for the ellipses who do not have their centers as their point of origin is the fact that they are required to have a strong knowledge of the genre of trigonometry. This is important since the individuals are required to take the help of the different components of trigonometry like cos, tan, sin and others for solving the equations related to the same. This might come as a surprise to many individuals since the above given equation looks rather simple which might propel the individuals to think that they have the necessary tools to solve it easily. However, the major problem arises because of the fact that without having an effective working knowledge of the subject of trigonometry it becomes almost impossible to solve the equation under discussion here. As a matter of fact, without the application of the different trigonometric formulas it is almost impossible to solve the above given equation.

An interesting trend seen within the framework of the contemporary mathematical world is the fact that the mathematicians are increasingly resorting to usage of different branches of knowledge for the purpose of improving the equations which had been used until now for the analysis of the different kinds of ellipses. As a matter of fact, it had been seen different new forms of older theorems have already been created or for that matter formulated and even attempts are being used to take the help of calculus for the purpose of solving equations related to the geometrical shape under discussion here. This attempt on the part of the mathematicians is important since it will not only help them to understand the nature of the ellipses but in turn the exact nature of the entire universe itself. This can be explained on the basis of the fact that the majority of the objects in nature are either elliptical in physical form or follow an elliptical path while in motion. Thus, by developing their knowledge regarding the geometrical shape of ellipses the mathematicians would be able to learn many new things regarding the entire universe itself which in turn is likely to improve the condition of the entire human race itself.  

The above discussion makes it apparent that although the circles and the ellipses look almost identical in nature with only a slight difference in the shape of the two yet a detailed analysis of the two clearly reveals the fact that both of them are very distinct from one another. As a matter of fact, the above discussion makes it very clear that the formulas which are applicable in the particular context of the circles do not hold true in the particular context of the ellipses as well. In this regard, it needs to be said that it is not possible to get the desired results from the ellipses through the usage of the formulas which are applicable in the particular context of the circles because of the inherent differences between the two. More importantly, the equations for the calculation of the foci, axis, centre and others are completely different in the particular of the two shapes under discussion here. Thus, it would be safe to say that although an ellipse is a special case of the circle yet the two are poles apart from one another be in terms of the properties or the characteristics that they display or for that matter the equations which are being used for the analysis of the two.

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