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I have read the Kepler's laws of motion where I came across the term foci of elliptical path. But what is really a focus? Can anyone please explain what is a focus (maybe with a diagram) and why a ellipse has two foci while a circle has one?

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    $\begingroup$ "foci" is the plural of "focus". $\endgroup$
    – DanielSank
    May 23, 2015 at 7:24
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    $\begingroup$ Wouldn't this be better suited for the Maths SE? $\endgroup$ May 23, 2015 at 8:40

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The foci are simply points that define the ellipse by the relation $c^2 = a^2 - b^2$, where $c$ equals the length of each one of the foci to the center and $a$ is the length of a focus to the end of the ellipse.

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For a circle, $a$ = $b$.

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Given any two foci, a point on the ellipse is a point that is equal I he sum of he lengths of the foci.

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  • $\begingroup$ @Jimmy360-Is 'a' the length of a focus to the end of the ellipse or center to end of ellipse(as in first diagram)? $\endgroup$
    – Harmonic
    May 23, 2015 at 8:14
  • $\begingroup$ @soham Does this gif help? $\endgroup$
    – Jimmy360
    May 23, 2015 at 8:25
  • $\begingroup$ @Jimmy360-That's ok but is 'a' the length of focus to end of ellipse or is it length from center to end of ellipse? $\endgroup$
    – Harmonic
    May 23, 2015 at 13:04
  • $\begingroup$ @Jimmy360-Can you explain why length of 'a'(focus to end of the ellipse)is equal to length of focus to the co-vertex?do you mean that co-vertex as the end of the ellipse? $\endgroup$
    – Harmonic
    May 23, 2015 at 13:13
  • $\begingroup$ @soham a is the length of the half he major axis (the end to the center). I edited to add an important defining factor for an ellipse. $\endgroup$
    – Jimmy360
    May 23, 2015 at 14:26
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The equation of an ellipse whose semi-major and semi-minor axes are parallel to the x and y axes is given by: $$(\frac{x-h}{a})^2+(\frac{y-k}{b})^2=1$$ (where $a$,$b$ are the lengths of its semi-major and semi-minor axes respectively.)

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A focus, $c$ is defined by $c^2=a^2-b^2$, and therefore there can be two foci at a distance of $(a^2-b^2)^{1/2}$ on both sides of the centre, on the major axis. The equation of a circle is obtained when $a=b$. i.e. the semi-major and semi-minor axes are equal. (When $a=b$, $c=0$ and therefore the focus lies only at the center of the circle.)

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A circle is a degenerate ellipse, and you can also think of a circle as having two foci (on top of one another) as the eccentricity approaches 1. The foci of conic sections in general originate from the approach in which the curves are defined - using a focus (point) and directrix (line). This approach leads to rational parametric expressions for the conic sections. Of course there are other approaches to defining the curves.

A better question for the physics forum might be "why do celestial objects orbit in an ellipse?"

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