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Ellipse

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In geometry, we study about the conic section which are the shapes made by intersection of a cone and plane. There are four shapes that are obtained through this intersection - parabola, hyperbola ellipse and circle. An ellipse is very common conic section. It is formed when a cone is intersected by a slant plane.

Have a look at the diagram below :

Ellipse
They are common seen in engineering, physics and in astronomy as well. For instance - in our solar system, the orbits of plants are in elliptical shape. An ellipses are a kind of conic section that is a closed and bounded figure. The ellipses shape many properties with hyperbolas and parabolas which are another type of conic section but they are unbounded and open. The ellipse can also be thought to be formed by an intersection of a cylinder and a slant plane.

In this article, we are going to learn about the ellipses in detail. We shall discuss about the definition and equation of ellipse, about foci, eccentricity, area, and perimeter of the ellipse.

Definition

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An ellipse is a two dimensional curve that can be defined as the locus of a point sum of whose distances from two fixed points is a constant. It is a collection of special points in such a way that the ratio of distances of any point located on curve from any of the two fixed points to its distances from any of the two fixed lines is always the same and it called eccentricity. The pair of fixed points are foci (singular is focus), whereas the pair of fixed line is called directrices (singular is directrix). The shape of an ellipse resembles a circle which is compressed from one side.

Have a look at the following diagram:

Major Axis of Ellipse

An ellipse contains two axes inside it. These axes are perpendicular to each other about which the ellipse is supposed to be symmetric. The point where these two axes intersect, is termed as center point of an ellipse. The axis that is larger is termed as major axis and eventually the smaller one is called minor axis. The halves of major and minor axes are represented by the letter a and b. They are known as semi-major axis and semi-minor axis respectively. The point of intersections of elliptical curve and major axis represent two vertices.

There are two types of ellipses - horizontal and vertical. A horizontal ellipse is demonstrated by the image below:
Horizontal Ellipse
On the other hand, a vertical ellipse is explained by the diagram as under:
Vertical Ellipse

Equation and Formula

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Horizontal Ellipse:

The equations of horizontal ellipse are:

When center is at origin:

$\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1

When center is not origin:

Let the center of such ellipse be (h, k), then

$\frac{(x-h)^{2}}{a^{2}}$ + $\frac{(y-k)^{2}}{b^{2}}$ = 1

Vertical Ellipse:

The equations of vertical ellipse are:

When center is at origin:

$\frac{x^{2}}{b^{2}}$$\frac{y^{2}}{a^{2}}$ = 1

When center is not origin:

Let the center be (h, k),

$\frac{(x-h)^{2}}{b^{2}}$$\frac{(y-k)^{2}}{a^{2}}$ = 1

The term "foci" denotes the plural of focus. There are two focal points in an ellipse which are represented by $F_{1}$ and $F_{2}$. The foci of an ellipse are said to be two special points situated on the major axis of the ellipse. These points are at same distance from center on both sides. The coordinates of the foci are (c, 0) (right side focus) and (-c, 0) (left side focus).

Have a look at the following diagram:

Foci of Ellipse
Let us suppose that P be a point anywhere on the curve of ellipse. From this point, the sum of of distances of two foci is said to be constant which is equal to the length of major axis of the ellipse. i.e.

PF$_{1}$ + PF$_{2}$ = 2a

Eccentricity

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The eccentricity of an ellipse is a special ratio. It is defined as the ratio of the distance between the foci of ellipse to the major axis length. The eccentricity is denoted by letter e or sometimes by $\epsilon$. i.e.

e = $\frac{2f}{2a}$ = $\frac{f}{a}$

Where, f is the distance of each focus from center.

The eccentricity of an ellipse lies between 0 to 1 i.e. 0 < e < 1. We can say that when e = 0, both foci lie on the center and we obtain a circle. Eventually, the ellipse gets elongated shape, as e approaches to 1.

The area of an ellipse can easily calculated when we have the lengths of semi major and semi minor axes. The formula for the area of ellipse is given by the following formula:

Area = $\pi$ ab

Where, a and b are the lengths of halves of the major and minor axes. These lengths can be easily determined if the equation of ellipse is given. We can convert the equation in the corresponding standard form (if it is not). These form have been discussed in the above section.

Perimeter

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Perimeter of an ellipse refers to the length of its total circumference. The perimeter can be measured when the equation of an ellipse is given. We may determine the values lengths of semi minor as well as semi major axes i.e. a and b.
 
Then, these values are to be substituted in the following formula of perimeter.

Perimeter of ellipse = $2 \pi$ $\sqrt{\frac{a^{2}+b^{2}}{2}}$