# The Ellipse

**Definition of Ellipse**

Ellipse is the locus of point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2*a*.

**General Equation of the Ellipse**

From the general equation of all conic sections, *A* and *C* are not equal but of the same sign. Thus, the general equation of the ellipse is *Ax*^{2} + *Cy*^{2} + *Dx* + *Ey* + *F* = 0 or

### Standard Equations of Ellipse

From the figure above,

$d_1 = \sqrt{(x + c)^2 + y^2}$ and

$d_2 = \sqrt{(x - c)^2 + y^2}$

From the definition above,

$d_1 + d_2 = 2a$

$\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a$

$\sqrt{(x + c)^2 + y^2} = 2a - \sqrt{(x - c)^2 + y^2}$

Square both sides

$(x + c)^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + [ \, (x - c)^2 + y^2 \, ]$

$x^2 + 2xc + c^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + x^2 - 2xc + c^2 + y^2$

$2xc = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} - 2xc$

$4a\sqrt{(x - c)^2 + y^2} = 4a^2 - 4xc$

$a\sqrt{(x - c)^2 + y^2} = a^2 - xc$

Square again both sides

$a^2 [ \, (x - c)^2 + y^2 \, ] = (a^2 - xc)^2$

$a^2 [ \, x^2 - 2xc + c^2 + y^2 \, ] = a^4 - 2a^2xc + x^2c^2$

$a^2x^2 - 2a^2xc + a^2c^2 + a^2y^2 = a^4 - 2a^2xc + x^2c^2$

$a^2x^2 + a^2c^2 + a^2y^2 = a^4 + x^2c^2$

$a^2x^2 - x^2c^2 + a^2y^2 = a^4 - a^2c^2$

$(a^2 - c^2)x^2 + a^2y^2 = a^2(a^2 - c^2)$

From triangle OV_{3}F_{2} (see figure above)

$c^2 + b^2 = a^2$

$a^2 - c^2 = b^2$

Thus,

$b^2x^2 + a^2y^2 = a^2b^2$

Divide both sides by *a*^{2}*b*^{2}

$\dfrac{b^2x^2}{a^2b^2} + \dfrac{a^2y^2}{a^2b^2} = \dfrac{a^2b^2}{a^2b^2}$

$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

The above equation is the standard equation of the ellipse with center at the origin and major axis on the *x*-axis as shown in the figure above. Below are the four standard equations of the ellipse. The first equation is the one we derived above.

**Ellipse with center at the origin**

*x*-axis.

*y*-axis.

**Ellipse with center at (**

*h*,*k*)*h*,

*k*) and major axis parallel to the

*x*-axis.

*h*,

*k*) and major axis parallel to the

*y*-axis.