•  
agra,ahmedabad,ajmer,akola,aligarh,ambala,amravati,amritsar,aurangabad,ayodhya,bangalore,bareilly,bathinda,bhagalpur,bhilai,bhiwani,bhopal,bhubaneswar,bikaner,bilaspur,bokaro,chandigarh,chennai,coimbatore,cuttack,dehradun,delhi ncr,dhanbad,dibrugarh,durgapur,faridabad,ferozpur,gandhinagar,gaya,ghaziabad,goa,gorakhpur,greater noida,gurugram,guwahati,gwalior,haldwani,haridwar,hisar,hyderabad,indore,jabalpur,jaipur,jalandhar,jammu,jamshedpur,jhansi,jodhpur,jorhat,kaithal,kanpur,karimnagar,karnal,kashipur,khammam,kharagpur,kochi,kolhapur,kolkata,kota,kottayam,kozhikode,kurnool,kurukshetra,latur,lucknow,ludhiana,madurai,mangaluru,mathura,meerut,moradabad,mumbai,muzaffarpur,mysore,nagpur,nanded,narnaul,nashik,nellore,noida,palwal,panchkula,panipat,pathankot,patiala,patna,prayagraj,puducherry,pune,raipur,rajahmundry,ranchi,rewa,rewari,rohtak,rudrapur,saharanpur,salem,secunderabad,silchar,siliguri,sirsa,solapur,sri-ganganagar,srinagar,surat,thrissur,tinsukia,tiruchirapalli,tirupati,trivandrum,udaipur,udhampur,ujjain,vadodara,vapi,varanasi,vellore,vijayawada,visakhapatnam,warangal,yamuna-nagar

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

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?

image

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.

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).

image

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


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

image


Fig: showing an ellipse with Focus and directrix

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

image
           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.

image

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  

image        

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 F2.

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.

image
image

Tracing of the Ellipse

Equation of an ellipse is image

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,
image

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

image

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

image
           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

image

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 image

Solution :

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

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

image
 

Now, by definition, image

Using distance formulas, we get,image

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

Solution :

image

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 image( 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.                 imageimage

 

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

image

Let the equation of ellipse be image

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

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.image

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.

 

 

Talk to Our Expert Request Call Back
Resend OTP Timer =
By submitting up, I agree to receive all the Whatsapp communication on my registered number and Aakash terms and conditions and privacy policy