# Ellipse in a Cartesian Plane

## Ellipse in a Cartesian Plane

Locus of points for which the sum of the distances from two fixed points called foci is constant.

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$$_{1}$$(hk) and F$$_{2}$$(+ 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, bk) 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, ck) and F$$_{2}$$(h,−ck)
The coordinates of its vertices on the transverse axis are: S$$_{1}$$(h, ak) and S$$_{2}$$(h,−ak)
The coordinates of its vertices on the conjugate axis are: S$$_{3}$$(bh, k) and S$$_{4}$$(−b + hk)

### 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))