The Ellipse and Its Standard Equation

What makes a curve an ellipse

Start with a fixed point $S$ (the focus), a fixed line $l$ (the directrix), and a fixed positive number $e < 1$ (the eccentricity). An ellipse is the path traced by a point $P$ that always keeps its distance from $S$ equal to $e$ times its distance from $l$:

where $M$ is the foot of the perpendicular from $P$ to the directrix. This is exactly the same focus-directrix recipe that defines a parabola ($e=1$) — an ellipse is simply the case where the point is pulled a little closer to the focus than to the line, at every position, so the curve closes up into an oval instead of running off to infinity.

Building the standard equation

To get a clean equation we choose axes wisely. Let $A$ and $A'$ be the two points where the curve crosses the line through $S$ perpendicular to the directrix — these split $\overline{SZ}$ (Z being the foot of the perpendicular from S to the directrix) internally and externally in the ratio $e:1$. Taking the midpoint $C$ of $AA'$ as the origin and the line $AA'$ as the x-axis, and writing $CA = CA' = a$, the focus-directrix condition $SP = e\,PM$ turns — after expanding $(x-ae)^2 + y^2 = e^2(x - a/e)^2$ and simplifying — into

Because $0 < e < 1$, the quantity $a^2(1-e^2)$ is a positive number, so we can name it $b^2$ for some $b>0$. This gives the standard form of the ellipse:

Since $1-e^2 < 1$, we always get $b < a$ in this setup — the axis carrying the foci is the longer one.

Reading off the shape

Setting $y=0$ gives $x = \pm a$, so the curve meets the x-axis at $A(a,0)$ and $A'(-a,0)$; setting $x=0$ gives $y=\pm b$, meeting the y-axis at $B(0,b)$ and $B'(0,-b)$. The whole curve is trapped inside the rectangle $x=\pm a,\ y=\pm b$, and it is symmetric about both axes (if $(x,y)$ satisfies the equation, so do $(-x,y)$, $(x,-y)$, $(-x,-y)$). The point $C$ where the axes of symmetry cross is the centre of the ellipse.

The longer chord $AA'$ (length $2a$) is the major axis; the shorter chord $BB'$ (length $2b$) is the minor axis — whichever variable has the larger denominator underneath it in the equation tells you which axis is major. If $a=b$ the equation becomes $x^2+y^2=a^2$, an ordinary circle — so a circle is just the special, perfectly round case of an ellipse.

Worked Example

Problem. An ellipse has equation $\dfrac{x^2}{36} + \dfrac{y^2}{25} = 1$. Identify which axis is the major axis, find the coordinates of the four points where the curve meets the axes, and state the lengths of the major and minor axes. …