# Is it possible for an ellipse to have only one focus?

• I
• parshyaa
In summary, the conversation discussed the concept of an ellipse and the confusion surrounding its two focal points. The definition of an ellipse was explained as the set of points in a plane where the sum of the distances from two fixed points are equal. This led to a discussion on the various forms and proofs of the equation for an ellipse, which ultimately concluded that an ellipse has two foci. The possibility of a single focus was also mentioned, but it was clarified that this only applies to a special case known as a circle.
parshyaa
i think there is only one focus of ellipse and we can draw a ellipse with a single focus. please correct my thinking.

How would such an ellipse look like? Did you draw one, and if yes, how?

There are ellipses where both focal points are at the same place, but this special case has a different name which is usually preferred.

fresh_42
parshyaa said:
i think there is only one focus of ellipse and we can draw a ellipse with a single focus. please correct my thinking.
Your thinking makes no sense. You may be thinking of a parabola. If you are really thinking of an ellipse, then you just need to study them more.

parshyaa
so please explain me why ellipse has two focus

parshyaa said:
so please explain me why ellipse has two focus
If you spend any time studying the math of the ellipse it is very obvious why it has to focii. It's all there in the math and it's not hard. I'm puzzled by your confusion. Have you used the equation of an ellipse to draw one?

there are two definitions of ellipse one is the sum of the distance of a point from two foci is constant and another one is related to eccentricity. i was confused with the proof of the equation of ellipse using second definition because they have used only one foci and one directrix, which made me confused that why there are two foci, but now i got it. thank you my friends. this is the reason why i love physics forum

parshyaa said:
there are two definitions of ellipse one is the sum of the distance of a point from two foci is constant and another one is related to eccentricity. i was confused with the proof of the equation of ellipse using second definition because they have used only one foci and one directrix, which made me confused that why there are two foci, but now i got it. thank you my friends. this is the reason why i love physics forum
Ah, I see your confusion now. Just keep in mind that it is sometimes true that you can use different forms to describe the same thing but if they are indeed describing the same thing, then they have to be equivalent.

parshyaa
There is no focal point involved in the definition via ##\frac{x^2}{a^2} + \frac{y^2}{b^2}=1##. You can calculate the two focal points based on that equation, but that needs some work.

parshyaa

Then we have SA = e AZ ... (1)

And SA' = e A'Z ... (2)

.·. A and A' lie on the ellipse.

Let AA' = 2a and take O the midpoint of AA' as origin. Let P(x, y) be any point on the ellipse referred to OA and OB as co-ordinate axis.
Then from figure it is evident that

AS = AO - OS = a - OS

AZ = OZ - OA = OZ - a

A'S = A'O + OS = a + OS

A'Z = OZ + OA' = OZ + a

Substituting these values in (1) and (2), we have

a - OS = e (OZ - a) ... (3)

a + OS = e (OZ + a) ... (4)

Adding (3) and (4), we get

2a = 2 e OZ

Or OZ = a/e ... (5)

Subtracting (3) from (4), we get

2 OS = 2ae => OS = ae ... (6)

.·. The directrix MZ is x = OZ = a/e and the co-ordinate of the focus S are (OS, 0) i.e. (ae, 0). Now as P(x, y) lies on the ellipse.

So we get

SP = e PM or SP2 = e2 PM2

(x - ae)2 + y2 = e2 [OZ - x co-ordinate of P]2

=> (x - ae)2 + y2 = e2 [a/e - x]2 = (a - ex)2 ... (7)

=> x2 + a2e2 - 2axe + y2 = a2 + e2x2 - 2aex

or x2/a2 + y2/a2(1-e2) = 1 [Dividing each term by a2 (1 - e2)]

or x2/a2 + y2/b2 = 1 where b2 = a2 (1 - e2)
in this proof they have used only one focus, i know that from both the focus equation will be same. but this made me confused to treat ellipse having only one true focus.

from this proof how can we say that ellipse has two foci

For the ellipse, it does not matter what you use in a proof. Also, you indirectly used the symmetry of the ellipse, and the other focus is symmetric to the first one.

parshyaa
yes right

parshyaa said:
i think there is only one focus of ellipse and we can draw a ellipse with a single focus. please correct my thinking.
The requirement of two foci comes from the definition. The set of points in a plane whose sum of distance from two fixed points are equal. Said better in the first paragraph of this article: https://en.wikipedia.org/wiki/Derivation_of_the_Cartesian_form_for_an_ellipse

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parshyaa
mfb said:
How would such an ellipse look like? Did you draw one, and if yes, how?

There are ellipses where both focal points are at the same place, but this special case has a different name which is usually preferred.
Now that this topic has be discussed rather well, we can state that the name mfb refers to is "circle".

symbolipoint, parshyaa and mfb

## 1. Why does an ellipse have two foci?

An ellipse is a geometric shape that is formed by the intersection of a cone and a plane. The two foci of an ellipse are important for understanding its shape and properties. The foci are located on the major axis of the ellipse, which is the longest diameter of the shape. They are equidistant from the center of the ellipse and help define its shape and symmetry.

## 2. What is the significance of the two foci in an ellipse?

The two foci in an ellipse play a crucial role in determining its shape and size. The distance between the two foci, known as the major axis, is a key factor in calculating the eccentricity of the ellipse. The eccentricity is a measure of how elongated or circular the shape is, and is equal to the distance between the foci divided by the length of the major axis.

## 3. How are the two foci related to the other elements of an ellipse?

The two foci are related to the other elements of an ellipse in several ways. They are located on the major axis, which also contains the longest and shortest diameters of the ellipse. The distance from each focus to any point on the ellipse is equal to the sum of the distances from the two foci. This property is known as the focus-directrix property and is used to define an ellipse geometrically.

## 4. Can an ellipse have more than two foci?

No, an ellipse can only have two foci. This is a defining characteristic of an ellipse and differentiates it from other geometric shapes, such as a circle or a parabola. If an ellipse were to have more than two foci, it would no longer be considered an ellipse.

## 5. How do the two foci affect the motion of an object in an ellipse?

The two foci play a crucial role in the motion of an object in an ellipse. An object placed at one focus of an ellipse will follow an elliptical path around the other focus, known as the center of force. This motion is governed by Kepler's laws of planetary motion and is used to explain the orbits of planets and other celestial bodies.

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