Ellipse-2





In this page 'Ellipse-2' we are going to see some properties of ellipse and examples.             

Major axis is the line segment that contains both foci and vertices of the ellipse. It passes through the center of the ellipse. The length of the major axis is 2a.

Minor axis is the line segment that passes through the centre. It is the shortest diameter of the ellipse. Minor axis is perpendicular to the major axis. The length of the minor axis is 2b.

Vertices are the points of intersection of major axis with the ellipse.

Co-vertices are the points of intersection of minor axis with the ellipse.

Example:

State the centre, vertices, co vertices, and foci of the ellipse

4x²+9y²+16x-36y+16=0. Also find the length of the both axes.

Solution:

 4x²+9y²+16x-36y+16=0

 Subtracting 16 on both sides, we get

 4x²+9y²+16x-36y  = -16

 4x²+16x+9y²-36y  = -16

 4(x²+4x+4)+9(y²-4y+4)-36-16 = -16

 4(x²+4x+4)+9(y²-4y+4) =  -16+36+16

 4(x+2)²+9(y-2)² =  36

 Dividing by 36 on both sides  ellipse-2

 (x+2)²/9 + (y-2)²/4   =  1

The larger denominator is 9 which is a². So the major axis is the horizontal axis. a=3 and b=2

 The length of the major axis is 2a = 2(3) =6

 The length of the minor axis is 2b = 2(2) =4

 The centre of the ellipse is  = (-2,2)

 To find the foci first we have to find the eccentricity.

 To find the eccentricity from 'a' and 'b'  when a>b is

    b²=a²(1-e²)  

     4  =9(1-e²)

     4/9 = 1-e²

     e²  = 1-4/9

          =  5/9

     e  =  √5/3

Foci with respect to the origin are (ae,0) and (-ae,0)

Foci with respect to the centre (-2,2) are (-2+√5,2) and (-2-√5,2).

The vertices are 3 units on either side of the centre. So the vertices are (-5,2) and (1,2)

The co-vertices are 2 units above and below the centre(-2,2). So the co-vertices are (-2,4) and (-2,0)

The centre is (-2,2), the foci are (-2+√5,2) and (-2-√5,2). The vertices are (-5,2) and (1,2). The co-vertices are (-2,4) and (-2,0).  The length of the major axis is 6, the length of the minor axis is 4.

The parents and teachers can guide the students to follow the example in this page 'Ellipse-2' step by step,  and guide them to do the practice problem on their own.  If you have any doubt, you can contact us through mail, we will help you to clear your doubts.     

Practice problem:

Find the centre,foci, vertices, co-vertices and length of the axes of the ellipse   4x²+25y²+8x+100y+100=0     

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