Ellipse

"Ellipse" is defined as the locus of a point which moves such that the ratio of its distance (Eccentricity) from a fixed point (Locus) and a fixed line (Directrix) is less than one i.e. a point moves such that its distance from a fixed point is always less than the distance from a fixed line, we get a different types of curve for one value of eccentricity, which are similar for all values of eccentricity less than one.

It can be called a generalization of a circle i.e. a circle may be called as a special type of ellipse that has both focal points at the same location. Thus curve looks like a circle but it is not exactly a circle. Rather it is more like the edges of an egg. And if we plot the movement of the Earth and other planets around the Sun, it is the same curve satisfying the above condition of eccentricity less than one. This beautiful curve has been named as "Ellipse".

Mathematically, an ellipse may be defined as the locus of points satisfying the equation

where:

x,y are the coordinates of any point on the ellipse,
a, b are the radius on the x and y axes respectively

 

In the figure below, F1 and F2 are the two foci and the directrix are represented by D1 and D2.

 

 Ellipse with F1 and F2 as focus

 

Parametric Equation of an Ellipse

In parametric form, we may define an ellipse as the locus of all points satisfying the given equation:

x = a cos t

y = b sin t

where:

x,y are the coordinates of any point on the ellipse,

a, b are the radius on the x and y axes respectively

t is the parameter which ranges from 0 to 2π radians

To know the trend of questions asked in the examination, you may look into the papers of Previous Years.

We now discuss some of the important terms:

Focus

As depicted in the above figure by F1 and F2, the distance from the center to either focus is f = ae, which can be expressed in terms of the major and minor radii:

 

 

Eccentricity

The eccentricity of the ellipse (commonly denoted as either e or ) is

 

 The Tangent to an Ellipse at a Given Point

Differentiating the equation for an ellipse with respect to

 

Hence the gradient at   is 

The equation of the tangent at is therefore given by:

This can be written as:

 

Now, since the point (x1, y1) lies on the ellipse, the equation of the tangent at this point becomes

 

The Normal To The Tangent At (x', y’)

The Normal at the point is the line perpendicular to the tangent and therefore its slope is

and its equation is:

 

 

 In this chapter we will discuss in detail the nature/properties of this beautiful and important curve. As you will see, the curve is symmetrical about two axes. We will study the standard form of ellipse where the X and y-axes will be taken as these axes. The main emphasis in this chapter should be on learning the properties of ellipse.

Ellipse is important from the perspective of scoring high in IIT JEE as it fetches 1-2 questions in most of the engineering examination. You are expected to do all the questions based on this to remain competitive in IIT JEE examination

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