What is eccentricity, and why should you study about it?
There are 4 different kinds of shapes that are parabola, circle, hyperbola and ellipse. Very few of them are related to conics. When we talk regarding conic sections, let’s go back to 350 B.C, when the Greek civilisation found distinctive shapes derived from conic. They sliced the double cone in some distinctive approaches and finally discovered four new shapes.
In this article, we will explore the definition of Eccentricity and its importance.
What is the distinction between these forms?
You may be thinking that the shape’s structure is different and correct, but how do you distinguish between them in the world of math? If you look closely, all look similar to circles. For instance, take the time to look at an ellipse.
If you extend a circle starting from abscissa or the ordinate, then you’ll end up with an ellipse. In short, an ellipse is a squeezing circle. In this way, most of the shapes are like circles. They aren’t circles since they’re distinct shapes and have distinct properties. This is why we are introducing the concept of eccentricity.
Definition of Eccentricity
What exactly is eccentricity? In various books, you’ll find the definition as “Eccentricity is the measure of opening on the branches of the forms”.
Let us simplify this for you to understand. The eccentricity of the curve shows how non-circular it is. The process of determining eccentricity is simple; however, you must understand the reasoning behind it. If we discover an eccentricity, we will always use the value related to the reference as a reference.
So, what shape need to consider as the reference?
The Greek mathematicians already did the difficult part. After all their calculation and study, they used the circle as their reference point. That means that whenever you find the eccentricity of any form, it measures the eccentricity of the shape using the circle as a reference. In other words, it is in determining how similar the shape is to the circle. If the value of eccentricity is low, it is a sign that your shape is more like a circle.
If we consider that the circle is the reference frame, what is the circle’s eccentric value? The circle’s eccentric value is zero. This means that if the circle’s eccentricity is zero, then it is an ideal circle.
But, if its eccentric is, for instance, 0.002, you can’t claim it’s an ellipse, but when you draw the shape on a graph, it’s going to be difficult to determine that the shape is either an ellipse or circle. This is why eccentricity is crucial.
The Eccentricity of Conic Sections
There are various conics, including an ellipse, parabola, hyperbola, and circle. The eccentricity of a conic is defined as the distance that extends from a point to the centre of it divided by the distance perpendicular from the point to the nearest directrix. The value of eccentricity is constant for all conics.
Eccentricity Definition
For any conic segment, a locus of a location exists in which distances between that point (focus) and to the directrix (directrix) can be found in a constant ratio. This ratio is referred to as eccentricity. The sign “e” denotes it.
Eccentricity Formula
The formula used to determine the eccentricity of a conic section is described as follows:
Eccentricity, e = c/a
Where,
c = distance from the middle to the focus
a = distance from the middle to the vertex
For any conic sector, the equation generally is of the quadratic form:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Eccentricity of Circle
A circle can be defined as the set of points on an area equidistant from a fixed point on the plane’s surface known as “centre”. “Ridus” is the term used to describe a “radius” that defines the distance from the centre to the points on the circle. If the centre of the circle is located at its origin, then it’s simple to calculate the equation for the circle. The equation for the circle is calculated using the conditions below.
If “r” is the radius and (h, K) be the middle of the circle with the help of the definition, we will get = r.
We are aware that the formula used to calculate the distance is
√[(x –h)2+( y–k)2]= r
Take Square both sides, and we can
(x –h)2+( y–k)2= r2
Therefore, the equation for the circle that has centre C(h, K) and the radius “r” is:
(x –h)2+( y–k)2= r2
Additionally, the eccentricity of the circle is equal to zero, i.e. e = 0.
Eccentricity of Parabola
A parabola can be defined as the collection of points P in which the distances to a fixed-line F (focus) within the plan are the same as their distances from a fixed-line l(directrix) within the plan. Also, the distance from the fixed point of the plane is a constant ratio that is equal to the distance to the fixed line in the plane.
The distorting factor of the parabola is equal to 1, i.e. e = 1.
The equation for a general parabola can be expressed by x2 = 4ay, and the eccentricity is written as 1.
Eccentricity of Ellipse
An ellipse can be described as the collection of points in a plane where all distances between two fixed points are equal. The distance to the fixed point of the plane is the same ratio but less than the distance from the fixed line in the plane.
Thus, the exponent of the ellipse is less than 1, i.e. e is.
Eccentricity of Hyperbola
Hyperbolas are the number of points in an area where the distances between 2 fixed locations are the same. Also, the distance to the fixed point of the plane has a fixed ratio that is greater than the distance from the fixed-line within the plane.
Thus, the hyperbola’s eccentricity exceeds 1, i.e. e < 1.
Conclusion
So, that was all about eccentricity. As it’s not something that can be understood and remembered by reading it once, we’d recommend you study it until you’re confident that you can score good marks if eccentricity related questions are asked in the exams.
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