Ellipse is a member of the conic section and has features similar to a circle. An ellipse, unlike a circle, has an oval shape. The locus of points is represented by an ellipse with an eccentricity less than one, and the total of their distances from the ellipse's two foci is a constant value. The shape of an egg in two dimensions and the running track in a sports stadium are two simple examples of the ellipse in our daily lives.

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The ellipse is a portion of the conic segment, which is the point where a cone meets a plane that does not cross the base of the cone. The focus is designated by S, the constant ratio 'e' is known as the eccentricity, and the fixed-line is known as the directrix (d) of the ellipse.

The length of an ellipse's boundary is its perimeter. It's also known as the ellipse's circumference. The length of the continuous line that forms the ellipse's boundary is its perimeter. Unfortunately, unlike other forms, there is no method for calculating an ellipse's exact (or) accurate perimeter. However, there are numerous approximation formulas for calculating the approximate perimeter value, such as:

1. Formulas for approximation

2. Formulas of Ramanujan

3. Formulas that make use of infinite series

4. Formulas that incorporate integration

Ellipse Components

Let's go through a few keywords related to the various sections of an ellipse.

• Focus: F(c, o), and F' are the coordinates of the two foci on the ellipse (-c, 0). As a result, the distance between the foci is equal to 2c.

• Centre: The centre of the ellipse is the middle of the line connecting the two foci.

• Major Axis: The length of the ellipse's major axis is 2a units, and the end vertices of this major axis are (a, 0), (-a, 0), respectively.

• Minor Axis: The length of the ellipse's minor axis is 2b units, and the end vertices of the minor axis are (0, b), and (0, -b), respectively.

• Latus Rectum: The latus rectum is a line that is drawn perpendicular to the ellipse's transverse axis and passes through the ellipse's foci. 2b2/ais the length of the ellipse's latus rectum.

• Transverse Axis: The transverse axis is the line that connects the two foci and the ellipse's centre.

• Conjugate Axis: The conjugate axis is a line that passes through the centre of the ellipse and is perpendicular to the transverse axis.

• Eccentricity: The distance of the focus from the ellipse's centre is divided by the distance of one end of the ellipse from the ellipse's centre. If the focus distance from the ellipse's centre is 'c,' and the end distance from the centre is 'a', eccentricity e = c/a.

Properties of Ellipse 

There are a number of characteristics that distinguish an ellipse from other comparable shapes. These are the properties of an ellipse:

1. An ellipse is formed when a plane intersects a cone at its base angle.

2. There are two foci or focal points in every ellipse. The distances between any point on the ellipse and the two focus points add up to a constant value.

3. All ellipses have a centre as well as a major and minor axis.

4. All ellipses have an eccentricity value of less than one.

How do you find the Ellipse's Area?

The following are the processes for calculating the area of an ellipse using the lengths of the major and minor axes:

Step 1: Determine the distance between the ellipse's farthest point and the centre ('a', or the length of the semi-major axis).

Step 2: Calculate the distance between the centre and the closest point on the ellipse ('b,' or the length of the semi-minor axis).

Step 3: Take the product of a and b and multiply it by.

Step 4: Calculate the area in square metres.

Important Reminders

1. Ellipse is the locus of all points on a plane whose sum of distances between two fixed points is constant.

2. Area of ellipse = π a b

3. Assume that the value of π is 3.14 or 22/7.

4. Use the formula for the area of an ellipse to find the values of the semi-major axis (a) and the semi-minor axis (b).

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Ellipse Formula

As we know, we can define an ellipse as a closed-shape structure in a two-dimensional plane. Hence, the ellipse covers a region in a 2D plane. So, this bounded region of the ellipse is the area of the ellipse.  The shape of the ellipse is different from that of the circle, hence the formula for its area will also be different.

Let’s discuss the area and the perimeter of the ellipse.

Area of Ellipse

The area of the circle is calculated based on its radius, but the area of the ellipse depends on the length of the minor axis and major axis.

Area of the circle = πr²

And,

Area of the ellipse = Pie(π) x Semi-Major Axis x Semi-Minor Axis

Perimeter of Ellipse

The perimeter of an ellipse can be defined as the total distance run by its outer boundary. For a circle, it is very easy to find its circumference, since the distance from the centre to any point of locus of a circle is the same. This distance is called the radius.

But in the case of an ellipse, we have two axes, the major and minor axis, that cross through the centre and intersect. Hence, an approximation formula can be used to find the perimeter of an ellipse:

The perimeter of Ellipse = 2 \[\pi \sqrt{\frac{a^{2}+b^{2}}{2}}\]

Where a is the length of the semi-major axis and b is the length of the semi-minor axis respectively.

What is Latus Rectum?

The line segments that are perpendicular to the major axis through any of the foci such that their endpoints lie on the ellipse are defined as the latus rectum.

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The length of the latus rectum is \[\frac{2b^{2}}{a}\].

L = \[\frac{2b^{2}}{a}\]

Where a is the length of the minor axis and where b is the length of the major axis.

Solved Examples

Question 1. If the length of the semi-major axis is  8 cm and the semi-minor axis is 5cm of an ellipse. Find its area.

Answer:  Given, length of the semi-major axis of an ellipse, a = 8cm

length of the semi-minor axis of an ellipse, then b equals 5cm

By the formula of area of an ellipse, we know that;

Area = π × a × b

Area = π × 8 × 5

Area = 40 π

or

Area = 40 × 22/7

Area = 110 cm²

Question 2. If the length of the semi-major axis is 10 cm and the semi-minor axis is 5 cm of an ellipse. Find the perimeter of the ellipse.

Answer: Given, length of the semi-major axis of an ellipse, a = 10 cm

length of the semi-minor axis of an ellipse, b equals 5cm

By the formula of Perimeter of an ellipse, we know that;

The perimeter of ellipse = 2 \[\pi \sqrt{\frac{a^{2}+b^{2}}{2}}\]

Therefore, the Perimeter of ellipse = 2×3.14 \[\frac{10^{2}+5^{2}}{2}\] = 49.64

Fun Facts

1. The ellipse was first studied by Menaechmus, investigated by Euclid, and was named by Apollonius. The focus and conic section directrix of an ellipse was considered by Pappus. In 1602, it was Kepler who believed that the orbit of Mars was oval; he later discovered that it was an ellipse with the Sun at one focus.

2. An ellipse and a circle are both known to be examples of conic sections.

3. A circle is known as a special case of an ellipse, with the same radius for all points.

4. By stretching a circle in the x-direction or y-direction, an ellipse is created.

5. An ellipse is formed when a plane intersects a cone at an angle to its base.

6. All ellipses have two focal points or foci. The sum of the distances from every point on the ellipse to the two foci is known to be constant.

7. All ellipses have a centre, a major axis, and a minor axis.

8. All ellipses have eccentricity values greater than zero or equal to zero, and less than one.

9. Real-Life Examples of Ellipse Many real-world situations can be represented by ellipses, including orbits of various planets, satellites, moons, and comets, and shapes of boat keels, rudders, and some aeroplane wings.