Dimension of the Ellipse

There coronavirus ships in geometry such as the square, rectangle, circle, ellipse, etc. All these shapes have different properties and have different equations to represent them. Ellipse is one of the important shapes in geometry. The shape of our earth is elliptical. 

The ellipse is a plan cow that is surrounded by two focal points. The sum of both the distances to the focal point is constant. The ratio of the distance to the focal points and the line, which is the directrix of an ellipse, is called eccentricity. This ratio is always constant.

Ellipse

Ellipse is a planar curve that is surrounded by two focal points such a way that for every point on the curve, the distance of point 21 focus plus the distance of a point to the other focus is constant.

The area of the ellipse can be obtained by algebraic expressions; however, the perimeter of the ellipse can only be determined approximately.

The eccentricity of an ellipse is always less than 1.

Properties of Ellipse:

• A locus of points fulfilling certain conditions is represented by the ellipse

• A circle is also an ellipse in which both of the foci coincide with each other

• An ellipse can be represented with the help of the ellipse equation

• The equation of ellipse comes in various forms

• It helps identify the eccentricity of the ellipse along with other components

Let’s take a look at the components of an ellipse.

Components of an Ellipse

• The two foci of the Ellipse:

The ellipse has two foci, F1(c, o) and F2( -c, o); the distance between the two foci of the ellipse is equal to 2c.

Centre: There is a line that joins the two foci of the ellipse. The midpoint of this line is regarded as the centre of the ellipse.

• Eccentricity: 

The distance of focus from the centre of an ellipse and distance from the centre of the ellipse to one end of the ellipse from a ratio. This ratio is called the eccentricity of the ellipse.

• The Latus Rectum:

The transverse axis of an ellipse has a line perpendicular drawn to it. It passes through both foci of the ellipse. This line is known as the latus rectum of the ellipse. 2b²/a is the length of the lattice rectum of an ellipse.

The Eccentricity of an Ellipse

As discussed above, the eccentricity of an ellipse is a ratio. It is the ratio of the distance between the centre and focus of the ellipse and the centre of the ellipse to one end of the ellipse.

The eccentricity of an ellipse is represented by the symbol e.

The numerical value of the eccentricity of an ellipse is always less than 1.

The formula for the eccentricity of the ellipse is given below:

Eccentricity = e = c/a = √( 1- (b2 – a2))

The eccentricity of an ellipse can also be defined as how much the conic section or the ellipse varies from being a circle.

It tells us how much an ellipse is closer to being a circle in the following ways.

The more the eccentricity, the more the ellipse is closer to becoming a circle.

Less the eccentricity, less the ellipse is closer to becoming a circle.

The Standard Equation of an Ellipse

The ellipse has two standard equations called the ellipse equations.

They are based on the transverse axis of the ellipse and the conjugate axis of the ellipse.

Let a sticker look at two standard equations of an ellipse.

(x2 / a2) + (y2 / b2) = 1

In this standard equation of an ellipse, the transverse axis of the ellipse is the x-axis, and the y-axis is the conjugate axis of the ellipse.

(x2 / b2) + (y2 / a2) = 1

In this form of the standard equation, the y axis is the transverse axis of the ellipse, and the x-axis is the conjugate axis of the ellipse.

Let us take an example to find the equation of an ellipse that has a major axis and passes through the points (-3, 1) and (2, -2).

According to the standard form of the ellipse: 

 (x2 / a2) + (y2 / b2) = 1

Now let us substitute the values,

(9 / a2) + (1 / b2) = 1

and 

(4 / a2) + (4 / b2) = 1

Now solve both these equations simultaneously, and you will get the following values.

a2= 32 / 3

and

b2 = 32 / 5

Hence, the equation of an ellipse becomes

(x2 / (32 / 3) + (y2 / (32 / 5) ) = 1

So, the final standard equation of the ellipse that passes through the points (2, -2) and (-3, 1) is

3x² + 5y2 = 32.

Conclusion

An ellipse is a geometrical shape that can be represented in the form of algebraic expression. An ellipse has two foci. The addition of distance of any point on an ellipse with both of the foci is always constant. A circle is a special form of ellipse where both foci of the ellipse coincide.

Eccentricity is the property of an ellipse which determines how close an ellipse is to be a circle. The eccentricity of an ellipse will always be less than one. The ellipse can be represented in the standard equation. There are two types of standard equations for an ellipse.