You are watching: What is the maximum eccentricity an ellipse can have

**If an ellipse is close to circular it has actually an eccentricity close to zero. If an ellipse has an eccentricity close come one it has actually a high level of ovalness.Figure 1 mirrors a picture of two ellipses among which is practically circular with an eccentricity close come zero and the various other with a greater degree of eccentricity.**The formal definition of eccentricity is:

ECCENTRICITY OF an ELLIPSE:

The eccentricity (e) of an ellipse is the proportion of the distance from the center to the foci (c) and also the street from the center to the vertices (a).

e= c a

As the distance between the center and also the foci (c) ideologies zero, the ratio of c a viewpoints zero and the shape approaches a circle. A circle has actually eccentricity same to zero.As the distance in between the center and the foci (c) approaches the distance between the center and the vertices (a), the ratio of c a ideologies one. One ellipse through a high degree of ovalness has an eccentricity draw close one.Let"s use this ide in part examples:

Example 1: uncover the eccentricity the the ellipse x 2 9 + y 2 16 =1

| a 2 =16→a=4
b 2 =9→b=3 c 2 = 4 2 − 3 2 → c 2 =7→c= 7 |

action 2: substitute the values for c and also a into the equation because that eccentricity. | e= c a e= 7 4 →e≈0.66 |

Example 2: discover the typical equation that the ellipse through vertices at (4, 2) and (-6, 2) v an eccentricity of 4 5 .

➢ the collaborates of the facility (h, k). ➢ the size of half the major axis (a). ➢ the street of half the boy axis (b). See more: Does The Curtain Match The Drapes Meaning, What Does The Carpet Matches The Drapes Mean | Orientation of significant axis: due to the fact that the 2 vertices fall on the horizontal heat y = 2, the significant axis is horizontal.
( h, k )=( 4+( −6 ) 2 , 2+2 2 )=( − 2 2 , 4 2 )=( −1,2 )
vertex (4, 2): c=| 4−( −1 ) |=| 5 |=5 vertex (-6, 2): c=| −6− ( −1 ) |=| −5 |=5 a = 5
e= 4 5 = c a 4 5 = c 5 →20=5c→c=4 c 2 = a 2 − b 2 →b= a 2 − c 2 b= 5 2 − 4 2 →b= 9 →b=3 |

step 2: instead of the worths for h, k, a and also b right into the equation for an ellipse through a horizontal significant axis. | Horizontal significant axis equation: ( x−h ) 2 a 2 + ( y−k ) 2 b 2 substitute values: < x−( −1 ) > 2 5 2 + ( y−2 ) 2 3 2 =1 Simplify: ( x+1 ) 2 5 2 + ( y−1 ) 2 3 2 =1 |

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