Every ellipse can be changed back, through a translation or a rotation, to a

**standard ellipse**whose centre is at the origin of the Cartesian plane and whose foci are on the

*x*-axis or on the

*y*‑axis.

### Equations

The standard ellipse is the ellipse centered at the origin, whose vertices are \(S_1(a, 0)\) and \(S_2(−a, 0)\) on the *x*-axis and \(S_3(0, b)\) and \(S_4(0, −b)\) on the *y*-axis; its foci are the points with coordinates \(F_1(c, 0)\) and \(F_2(−c, 0)\).

If the ellipse is centred at the origin, the equations are the following:

- \(\dfrac{x^2}{a^2}\) + \(\dfrac{y^2}{b^2}\) = 1, where \({c^2}\) = \({a^2}\) – \({b^2}\) if the transverse axis is the
*x*-axis:

The coordinates of the centre of the ellipse are: C(0, 0)

The coordinates of its foci are: F\(_{1}\)(*c*, 0) and F\(_{2}\)(−*c*, 0)

The coordinates of its vertices on the transverse axis are: S\(_{1}\)(*a*, 0) and S\(_{2}\)(−*a*, 0)

The coordinates of its vertices on the conjugate axis are: S\(_{3}\)(0,* b*) and S\(_{4}\)(0,−*b*)

- \(\dfrac{y^2}{a^2}\) + \(\dfrac{x^2}{b^2}\) = 1, where \({c^2}\) = \({a^2}\) – \({b^2}\), if the transverse axis is the
*y*-axis:

The coordinates of the centre of the ellipse are: C(0, 0)

The coordinates of its foci are: F\(_{1}\)(0,* c*) and F\(_{2}\)(0, −*c*)

The coordinates of its vertices on the transverse axis are: S\(_{1}\)(0, *a*) and S\(_{2}\)(0, −*a*)

The coordinates of its vertices on the conjugate axis are: S\(_{3}\)(*b*, 0) and S\(_{4}\)(−*b*, 0)

If the ellipse is not centred at the origin, the following data is obtained:

- \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\), for an ellipse whose transverse axis is parallel to the
*x*-axis:

The coordinates of the centre of the ellipse are: C(h,k).

The coordinates of its foci are: F\(_{1}\)(*c *+ *h*, *k*) and F\(_{2}\)(−*c *+ *h*, *k*).

The coordinates of its vertices on the transverse axis are: S\(_{1}\)(*a* + *h*, *k*) and S\(_{2}\)(−*a* + *h*, *k*).

The coordinates of its vertices on the conjugate axis are: S\(_{3}\)(*h*, *b* + *k*) and S\(_{4}\)(*h*,−*b* + *k*).

- \(\frac{(y-k)^{2}}{a^{2}}+\frac{(x-h)^{2}}{b^{2}}=1\), for an ellipse whose transverse axis is parallel to the
*y*-axis:

The coordinates of its foci are: C(*h*, *k*)

The coordinates of its foci are: F\(_{1}\)(*h*, *c* + *k*) and F\(_{2}\)(*h*,−*c* + *k*)

The coordinates of its vertices on the transverse axis are: S\(_{1}\)(*h*, *a* + *k*) and S\(_{2}\)(*h*,−*a* + *k*)

The coordinates of its vertices on the conjugate axis are: S\(_{3}\)(*b* + *h*, *k*) and S\(_{4}\)(−*b + **h*, *k)*

### Example

In the illustration below, the standard ellipse was translated 6 units to the right and 3 units downward: (6, −3). Therefore, the following can be noted:

The coordinates of the centre of the ellipse are: C(6, −3)

The coordinates of the foci are: F\(_{1}\)(4 + 6, −3) and F\(_{2}\)(−4 + 6, −3)

The coordinates of its vertices on the transverse axis are: S\(_{1}\)(5 + 6, −3) and S\(_{2}\)(−5 + 6, −3)

The coordinates of its vertices on the conjugate axis are: S\(_{3}\)(6, 3 −3) and S\(_{4}\)(6, −3 + (−3))