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In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum ${\displaystyle \ell }$, as follows:

${\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\\ell &=a(1-e^{2}),\\a\ell &=b^{2}.\end{aligned}}}$

The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola.

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping ${\displaystyle \ell }$ fixed. Thus a and b tend to infinity, a faster than b.

The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola.

Ellipse

The equation of an ellipse is

${\displaystyle {\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1,}$

where (h, k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x, y).

The semi-major axis is the mean value of the maximum and minimum distances ${\displaystyle r_{\text{max}}}$ and ${\displaystyle r_{\text{min}}}$ of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis

${\displaystyle a={\frac {r_{\text{max}}+r_{\text{min}}}{2}}.}$

In astronomy these extreme points are called apsides.^[1]

The semi-minor axis of an ellipse is the geometric mean of these distances:

${\displaystyle b={\sqrt {r_{\text{max}}r_{\text{min}}}}.}$

The eccentricity of an ellipse is defined as

${\displaystyle e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}},}$

so

${\displaystyle r_{\text{min}}=a(1-e),\quad r_{\text{max}}=a(1+e).}$

Now consider the equation in polar coordinates, with one focus at the origin and the other on the ${\displaystyle \theta =\pi }$ direction:

${\displaystyle r(1+e\cos \theta )=\ell .}$

The mean value of ${\displaystyle r=\ell /(1-e)}$ and ${\displaystyle r=\ell /(1+e)}$, for ${\displaystyle \theta =\pi }$ and ${\displaystyle \theta =0}$ is

${\displaystyle a={\frac {\ell }{1-e^{2}}}.}$

In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.

The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.

The semi-minor axis b is related to the semi-major axis a through the eccentricity e and the semi-latus rectum ${\displaystyle \ell }$, as follows:

${\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\a\ell &=b^{2}.\end{aligned}}}$

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping ${\displaystyle \ell }$ fixed. Thus a and b tend to infinity, a faster than b.

The length of the semi-minor axis could also be found using the following formula:^[2]

${\displaystyle 2b={\sqrt {(p+q)^{2}-f^{2}}},}$

where f is the distance between the foci, p and q are the distances from each focus to any point in the ellipse.

Hyperbola

The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches; if this is a in the x-direction the equation is:^{[citation needed]}

${\displaystyle {\frac {\left(x-h\right)^{2}}{a^{2}}}-{\frac {\left(y-k\right)^{2}}{b^{2}}}=1.}$

In terms of the semi-latus rectum and the eccentricity, we have

${\displaystyle a={\ell \over e^{2}-1}.}$

The transverse axis of a hyperbola coincides with the major axis.^[3]

In a hyperbola, a conjugate axis or minor axis of length ${\displaystyle 2b}$, corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints ${\displaystyle (0,\pm b)}$ of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length b. Denoting the semi-major axis length (distance from the center to a vertex) as a, the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows:

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1.}$

The semi-minor axis is also the distance from one of focuses of the hyperbola to an asymptote. Often called the impact parameter, this is important in physics and astronomy, and measure the distance a particle will miss the focus by if its journey is unperturbed by the body at the focus.^{[citation needed]}

The semi-minor axis and the semi-major axis are related through the eccentricity, as follows:

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