Writing Equations of Ellipses in Standard Form

To derive the equation of an ellipse centered at the origin, we begin with the foci $(−c,0)$ and $(c,0)$. The ellipse is the set of all points $(x,y)$ such that the sum of the distances from $(x,y)$ to the foci is constant, as shown in Figure 5.

Figure 5

If $(a, 0)$ is a vertex of the ellipse, the distance from $(-c, 0)$ to $(a, 0)$ is $a-(-c)=a+c$. The distance from $(c, 0)$ to $(a, 0)$ is $a-c$. The sum of the distances from the foci to the vertex is

$(a+c)+(a-c)=2 a$

If $(x, y)$ is a point on the ellipse, then we can define the following variables:

$\begin{aligned}&d_{1}=\text { the distance from }(-c, 0) \text { to }(x, y) \\&d_{2}=\text { the distance from }(c, 0) \text { to }(x, y)\end{aligned}$

By the definition of an ellipse, $d_{1}+d_{2}$ is constant for any point $(x, y)$ on the ellipse. We know that the sum of these distances is $2 a$ for the vertex $(a, 0)$. It follows that $d_{1}+d_{2}=2 a$ for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.

$\begin{array}{ll} d_{1}+d_{2}=\sqrt{(x-(-c))^{2}+(y-0)^{2}}+\sqrt{(x-c)^{2}+(y-0)^{2}}=2 a & \text { Distance formula } \\ \sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}=2 a & \text { Simplify expressions. } \\ \sqrt{(x+c)^{2}+y^{2}}=2 a-\sqrt{(x-c)^{2}+y^{2}} & \text { Move radical to opposite side. } \\ (x+c)^{2}+y^{2}=\left[2 a-\sqrt{(x-c)^{2}+y^{2}}\right]^{2} & \text { Square both sides. } \\ x^{2}+2 c x+c^{2}+y^{2}=4 a^{2}-4 a \sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2} & \text { Expand the squares. } \\ x^{2}+2 c x+c^{2}+y^{2}=4 a^{2}-4 a \sqrt{(x-c)^{2}+y^{2}}+x^{2}-2 c x+c^{2}+y^{2} & \text { Expand remaining squares. } \\ 2 c x=4 a^{2}-4 a \sqrt{(x-c)^{2}+y^{2}}-2 c x & \text { Combine like terms. } \\ 4 c x-4 a^{2}=-4 a \sqrt{(x-c)^{2}+y^{2}} & \text { Isolate the radical. } \\ c x-a^{2}=-a \sqrt{(x-c)^{2}+y^{2}} & \text { Divide by 4. } \\ {\left[c x-a^{2}\right]^{2}=a^{2}\left[\sqrt{(x-c)^{2}+y^{2}}\right]^{2}} & \text { Square both sides. } \\ c^{2} x^{2}-2 a^{2} c x+a^{4}=a^{2}\left(x^{2}-2 c x+c^{2}+y^{2}\right) & \text { Expand the squares. } \\ c^{2} x^{2}-2 a^{2} c x+a^{4}=a^{2} x^{2}-2 a^{2} c x+a^{2} c^{2}+a^{2} y^{2} & \text { Distribute } a^{2} . \\ a^{2} x^{2}-c^{2} x^{2}+a^{2} y^{2}=a^{4}-a^{2} c^{2} & \text { Rewrite. } \\ x^{2}\left(a^{2}-c^{2}\right)+a^{2} y^{2}=a^{2}\left(a^{2}-c^{2}\right) & \text { Factor common terms. } \\ x^{2} b^{2}+a^{2} y^{2}=a^{2} b^{2} & \text { set } b^{2} = a^2 - c^2 . \\ \frac{x^{2} b^{2}}{a^{2} b^{2}}+\frac{a^{2} y^{2}}{a^{2} b^{2}}=\frac{a^{2} b^{2}}{a^{2} b^{2}} & \text { Divide both sides by } a^{2} b^{2} . . \\ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 & \text { Simplify. } \end{array}$

Thus, the standard equation of an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$. This equation defines an ellipse centered at the origin. If $a > b$, the ellipse is stretched further in the horizontal direction, and if $b > a$, the ellipse is stretched further in the vertical direction.