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# Ellipse|Definition & Meaning

## Definition

A regular oval shape called an ellipse can be created by a point moving in a plane in a way that the summation of its distances from the foci remains constant or by cutting a cone with an oblique plane that does not cross its base.

**Ellipse** is a conic section component with properties similar to a** circle**. In contrast to a circle, an ellipse has an oval shape. An ellipse has an eccentricity below one and represents the locus of points whose distances from the **ellipse’s two foci** are a **constant value**. Ellipses can be found in our daily lives in a variety of places, including the two-dimensional shape of an egg and the running tracks in sporting venues.

## Equation of Ellipse

The general ellipse equation is used to represent an ellipse in the coordinate plane algebraically. An ellipse’s equation is as follows:

\[\frac { ( x-u )^2 } { a^2 } + \frac { ( y-v )^2 } { b^2 } = 1 \]

### Ellipse Standard Equation

The ellipse has two standard equations. These equations are built around the transverse and conjugate axes of each ellipse. The **transverse axis** is the **x-axis** in the standard ellipse equation $\frac {( x-u )^ 2 } { a ^ 2 } + \frac { ( y-v ) ^ 2 } { b ^ 2 } = 1 $ and the** conjugate axis is the y-axis**.

Furthermore, another standard ellipse equation is $\frac { x ^ 2 } { b ^ 2 } + \frac { y ^ 2 } { a ^ 2 } = 1 $, **with the transverse axis as the y-axis and the conjugate axis as the x-axis**. The image below depicts the two standard forms of ellipse equations.

## Ellipse Components

Let’s review some key terms related to the various parts of an ellipse.

**Focus:**The ellipse has two foci, which have the coordinates**F(c, o)**and**F’ (-c, 0)**. Thus, the distance between the**foci**is equal to**2c**.**Center:**The ellipse’s center is where the major and minor axes meet.**Major Axis:**The end vertices of the ellipse are (a, 0), and (-a, 0), with the major axis’ length being 2a units.**Minor Axis:**The end vertices of the ellipse are (0, b) and (0, -b), respectively, and the minor axis’ length is 2b units.**The Latus Rectum:**A line drawn perpendicular to the ellipse’s transverse axis is known as the latus rectum and passing through its**foci.**The ellipse’s latus rectum length is $ 2b^2 /a $.**Transverse Axis:**The axis that runs through the middle of the**ellipse**and between its two foci is known as the transverse axis.**Conjugate Axis:**The ellipse’s axis at a point equally spaced from the**foci**, which is perpendicular to the**transverse axis**.**Eccentricity (e<1):**A non-circular ellipse’s eccentricity is always greater than zero but less than one.

## Ellipse Characteristics

There are several characteristics that help distinguish an ellipse from other similar shapes. These ellipse properties are as follows:

- When a plane crosses a cone at its base angle, an ellipse is formed.
- Each ellipse has two focal points. Any two points on the ellipse have a fixed sum of their respective distances.
- There are major and minor axes on every ellipse, a centre, and eccentricity values that are less than one.

## How Do You Make an Ellipse?

There are specific steps to drawing an ellipse in math. The following is a step-by-step procedure for drawing an ellipse of given dimensions.

- Because the
**major axis**is the ellipse’s longest diameter, determine its length. - Draw one horizontal line the length of the major axis.
- Using a ruler, mark the
**midpoint**. The major axis length is divided by two to achieve this. - Using a compass, draw a
**circle of this diameter**. - Determine the length of the minor axis, which is the ellipse’s
**shortest diameter**. - Now, take the protractor and place it at the midpoint of the major axis. Make a mark at 90 degrees. Swing the protractor 180 degrees and mark the location.At its midpoint, the minor axis can now be drawn between the two marks.
- Using a compass, draw a circle of this diameter, just as we did for the major axis.
- Using a compass, divide the circle into
**twelve 30-degree sections**. - From the
**inner circle, make horizontal lines (but not for the major and minor axes).** - They run parallel to the
**main axis and radiate outward**from all intersections of the inner circle and the 30-degree line - Draw the lines a little shorter near the
**minor axis**and a little longer as you get closer to the main axis - From the
**outer circle**make vertical lines (but not for the minor and major axes). - These run parallel to the small axis and inward from all points where the outer circle and
**30-degree lines meet**. - Make the lines near the
**minor axis**a little longer. but a little shorter as you approach the main axis.If the horizontal line is too far, take a ruler and stretch it slightly before drawing the vertical line. - Use your best freehand drawing skills to draw the curves between the points.

## Ellipse Graph

Let’s look at a graphical representation of an ellipse using the ellipse formula. To graph an ellipse in a cartesian plane, certain steps must be taken.

### Step 1

Crossing with the coordinate axes, the ellipse intersects the x-axis at** A (a, 0), A'(-a, 0)**, and the y-axis at **B(0,b), B’ (0,-b)**.

**Step 2**

The ellipse’s vertices are **A(a, 0), A'(-a, 0), B(0,b),** and** B'(0,b) (0,-b)**.

**Step 3**

Because the ellipse is symmetric about the coordinate axes, It has two foci **S'(-ae, 0), S(ae, 0),** and two equation-based directories** d and d’** $ x = \frac { a } { e } $ and $ y = \frac {b } {e} $, respectively. Every chord is bisected by the origin** O**.As a result, origin O is the centre of the ellipse. As a result, it has the shape of a central conic.

**Step 4**

A closed curve is an ellipse that completely fills the rectangle.

**Step 5**

The major axis is the segment **AA′** of length **2a**, and the minor axis is the segment **BB′** of length **2b**. The major and minor axes are referred to as the ellipse’s principal axes.

## Example Ellipse

If the length of an ellipse’s semi-major axis is 10 cm and its semi-minor axis is 8 cm. Determine its location.

### Solution

Given the length of an ellipse’s semi-major axis, a = 10 cm, and the length of its semi-minor axis, b = 8 cm, we can calculate the area of an ellipse using the formula:

Area = $\pi$ x a x b

Area = $\boldsymbol\pi$ x 10 x 8 = 80 x $\pi$

**Area = 80 x 22/7 = 251.4286** **cm**$\mathsf{^2}$** area**

*All images/mathematical drawings were created with GeoGebra.*