Ellipse: Introduction to Ellipse, Definition, Equation , Tracing, Focus and Eccentricity

We know that all the planets including our earth revolve around the sun in a certain well-defined path/orbit. But do you know what is the name of the shape of this path?

The name given to such a shape is “Ellipse” and like every other shape, an ellipse will also have some important charateristics.So let’s try to understand this shape in detail.

Table of Contents

• Definition of Ellipse
• Standard equation of an ellipse
• Tracing of the ellipse
• Practice Problems
• FAQs

Definition of Ellipse

An ellipse is the locus of a moving point such that the ratio of its distance from a fixed point (focus) and a fixed line (directrix) is a constant. This constant distance is known as eccentricity (e) of an ellipse (0<e<1).

                      Fig: showing, fixed point,fixed line & a moving point

From definition of ellipse  Eccentricity (e) where 0<e<1

Fig: showing an ellipse with Focus and directrix

Note :
A second degree non-homogeneous equation,a,represents an ellipse if

           where  

Let A and B be two fixed points and P is a variable point in the plane. Then locus of P
will be an ellipse for the given condition i.e. PA+PB>AB. To satisfy this condition (triangular inequality), the point P should lie on the third vertex of the triangle ABP. Hence, the locus of the point P will be an ellipse as shown in the figure.

Another definition of ellipse: An ellipse can also be defined as the locus of a point whose sum of the distances from two fixed points (foci) is constant i.e. PF₁+PF₂=constant  

Standard Equation of an Ellipse

Let us consider a standard ellipse having foci F₁ (c,0) and F₂ (-c,0) and centre O(0,0) as shown in the figure.Let P be an arbitrary point on the ellipse. Connect point P to foci F₁ and F_2.

We know that, In a triangle, the sum of two sides is always greater than the third side.

⇒PF₁+PF₂>F₁ F₂

From the figure, F₁ F₂=c+c=2c  (distance between points F₁ and F₂  )

For an ellipse, PF₁+PF₂ will be some fixed value.

Tracing of the Ellipse

Equation of an ellipse is

The ellipse cuts the x-axis  at A(a,0)& A'(-a,0) and cuts the y-axis at B(b,0) & B'(-b,0)From the equation of ellipse,

Hence, ellipse is a closed curve lying entirely between the lines x=a  &  x= -a and the lines  y=b  &  y= -b

Practice Problems of Ellipse

Example : Does the equation 2x²+y²-4x-2y+5=0 represent an ellipse?

Solution :

A second degree non-homogeneous equation, ax²+by²+2hxy+2gx+2fy+c=0, represents an ellipse if △≠0 and h²-ab<0

           where  

Comparing the given equation with ax²+by²+2hxy+2gx+2fy+c=0,we get

a=2,b=1,h=2,g=-2,f=-1,c=5

And, h²-ab=(2)²-(1)(2)=2>0

The given equation does not represent an ellipse.

Example : Find the equation of the ellipse whose focus is (-1,1), directrix is x-y+3=0, and eccentricity is

Solution :

Given, Focus (F) ≡(-1,1) , Directrix: l ≡ x-y+3 = 0 and e =  

Let us consider P(h,k) be a moving point. 

Now, by definition,

Using distance formulas, we get,

Example : Check whether the equation  represents an ellipse or not.

Solution :

LHS represents ratio of distance of point (x,y) from the fixed point (2,5) and the fixed line 4x-3y+8=0 which is equal to ( i.e., between 0 and 1 )

So the given equation represents an ellipse.

Example : If the extremities of a line segment of length l move in two fixed perpendicular straight lines. Find the locus of that point which divides this lines segment in the ratio 1:2.

Solution :

Let the line segment be AB and the two fixed perpendicular straight lines be the co-ordinate axis and P(h,k) be the point whose locus is required as shown in the figure below.                 

Example : A tent is in the form of a semi ellipse. The base of which coincides with the road level. If the width of the road is 10m and a man 2m high just reaches the top when stands at 1m from one of the sides of the road. Find the greatest height of the arc.

Solution :

Drawing a figure using the information given in the question

Let the equation of ellipse be

From the figure, a=5. Hence, the equation of ellipse becomes  

The coordinates of the top of the person are P(4,2).Since, P lies on the curve, it will satisfy the equation of the curve.

FAQ’s of Ellipse

Question.1 Mention a few real life examples of an ellipse?
Answer: There are several real life examples of an ellipse like the shape of the orbit of planets, sommerfield elliptical orbit of electrons etc.

Question.2 Compare the eccentricities of Ellipse and Hyperbola.
Answer: The eccentricity of an ellipse is between 0 and 1 while the eccentricity of a hyperbola is greater than 1.

Question.3 Does the distance between foci change on shifting the center of an ellipse ?
Answer: The distance between foci always remains the same irrespective of the coordinates of its center.

Question.4 What does eccentricity tell us about an ellipse?
Answer: The eccentricity is the degree of flatness of an ellipse.It helps us understand how uncircular it is with reference to a circle.