Terms related to an ellipse,Standard form of Horizontal Ellipse,Practice Problems & FAQs

You must have heard that the earth revolves in an elliptical orbit with the sun at one of the foci. Apart from the foci there are certain more terminologies related to an ellipse such as the center, the vertices, the axes etc.

Let’s try to understand these terms in detail with the help of some examples.

Table of Contents

Terms related to an ellipse
The standard form of a horizontal ellipse
Practice Problems
FAQs

Terms related to an ellipse

Centre : The point that bisects every chord of an ellipse drawn through it is known as the centre of the ellipse.Here, centre

Foci : The foci are and or combined form of foci ≡

Vertices : The point of intersection of an ellipse with the line passing through the foci is known as vertices.Here, vertices are and .

Co-vertices : The point of intersection of an ellipse with the line perpendicular to the line passing through the foci is called co-vertices.

Here, co-vertices are and .

Eccentricity : As , Eccentricity,

Where, Distance between the centre and the focus

Distance between the centre and the vertex

We know that,

Also,

We know that, Length of the minor axis

Length of the major axis

Directrices : The equation of directrices are and or combined form of directrices,

Major axis : The line that passes through the foci and is perpendicular to the directrices is known as the major axis.

The major axis intersects the ellipse at the vertices.
The length of the major axis is the distance between its vertices, i.e.,
The major axis is the longest chord of an ellipse.
Half of the major axis i.e. or is the semi major axis of the ellipse. Its length is

Minor axis : The axis that is the perpendicular bisector of the major axis is known as the minor axis.

The minor axis intersects the ellipse at the co-vertices.
The length of the minor axis is the distance between its co-vertices, i.e.,
Half of the minor axis i.e. or is the semi minor axis of the ellipse. Its length is

Double ordinate : A chord perpendicular to the major axis is known as double ordinate.

There can be infinite double ordinates in an ellipse. From the figure, is the double ordinate.

Focal chord : A chord passing through the focus is known as focal chord.There can be infinite focal chords in an ellipse. From the figure, is the focal chord.

Latus rectum : The focal chord perpendicular to the major axis is known as latus rectum.

Here, latus rectum ➝ and

Equation of & equation of ⇒

Length of Latus rectum of the ellipse is given by .

Focal distance : The distance between the focus to any point on an ellipse is known as focal distance or focal radius.

Let be any point on the ellipse, then the focal distance is or .

As we know, by definition, and

⇒ and

We can see that ⇒

Also, ⇒

Therefore, focal distance of

Auxiliary Circle & Parametric equation of an ellipse

A circle described on the major axis as diameter is called an auxiliary circle.

For standard ellipse the equation of auxiliary circle is

Let be a point on the auxiliary circle.

Now, the coordinate of is .We can obtain the coordinate of by substituting the coordinate in the equation of ellipse.Hence the coordinates of are where is called eccentric angle of point

Hence, parametric equation of an ellipse is .

The standard form of a horizontal ellipse

Let and be the foci, and and be the directrixes of the ellipse.

Here, Centre of ellipse Origin or ,

Major axis of ellipse

Minor axis of ellipse

and

Equation of directrixes, and

Let be the eccentricity of the ellipse. The foci are and or the combined form of foci, ≡

The equation of directrixes are and or the combined form of directrixes,

By using this information, the complete diagram of a standard ellipse can be drawn as shown in the figure.

Practice Problems

Example : Consider the ellipse . Find its centre, vertices, eccentricity, foci, equation of the directrixes, length of the major axis, equation of the major axis, and length of the minor axis.

Solution :

Given, Equation of an ellipse

To convert this equation in the standard form, divide the whole expression by .

After comparing the given equation with , we get, and

and

We know that,

⇒

Now, we have, Centre

Vertices

Eccentricity

Foci

Equation of directrices,

Length of the major axis

Length of the minor axis

Example : If and are the foci of an ellipse passing through the origin, then find its eccentricity.

Solution :

Given, foci and the ellipse passes through the origin

Using, and

We get,

Now, we have to find the eccentricity.

So, or

Example : In an ellipse with the centre at the origin, if the difference of lengths of the major axis and the minor axis is 10 and one of the foci is at , then find the length of its latus rectum.

Solution :

Example : If the focal distance of an end of the minor axis of an ellipse (referred to its axis of

and respectively) is and the distance between its foci is , then find its equation.

Solution :

Given, distance between foci is .Let the equation of the ellipse be

We know that the distance between foci

⇒

And focal distance of one end of minor axis

We know,

So, the equation of ellipse is,

Example : If is any point on the ellipse where are the two foci then find the value of .

Solution :

We know that from any point on ellipse the sum of the distances from foci is i.e.

Given, we have

⇒

FAQs

Q 1. If both the foci of an ellipse coincides then what new geometrical shape would be obtained?
Answer: In case if both the foci of an ellipse coincides then we will obtain a Circle.

Q 2.Does the equation of ellipse have a term?
Answer: The equation of ellipse may have a term if its axes are not parallel to the coordinate axes.

Q 3.What is the difference between double ordinate and latus rectum?
Answer: The double ordinate is a chord perpendicular to the major axis of the ellipse whereas the latus rectum is a chord passing through focus and perpendicular to the major axis of the ellipse.