The Standard Equation of an Ellipse

The Standard Equation of an Ellipse — Moosa Academy

An ellipse is the set of all points whose distances to two fixed points — the foci — add up to the same constant. Its standard equation packs four pieces of information into a single line: where the centre sits, how wide it is, how tall it is, and which way it is stretched.

Concept The defining property

For every point

P

on the curve,

PF_1 + PF_2

is the same constant — and that constant equals

2a

, the full length of the major axis.

The centre is the midpoint between the foci, the vertices are the two points farthest from the centre, and the foci always lie on the major axis.

Concept The two standard forms

Horizontal major axis:

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Vertical major axis:

\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1

In both cases $(h,\ k)$ is the centre and $a > b$ . The single rule that decides the orientation is this: $a^2$ is always the larger denominator, and the major axis runs along whichever variable sits above it. If $a^2$ is under $x$ , the ellipse is stretched horizontally; if under $y$ , vertically.

Note What each element controls

Element	Meaning	Effect on the shape
h, k	Centre	Slides the whole ellipse; shape unchanged
a	Semi-major axis	Half the long width; larger a stretches it further
b	Semi-minor axis	Half the short width; as b nears a it rounds out
c	Centre-to-focus distance	Found from c² = a² − b²
e	Eccentricity, e = c/a	How flattened it is; 0 < e < 1 always

Note that $c^2 = a^2 - b^2$ uses a minus — the opposite of the hyperbola's plus. And when $a = b$ , $c$ becomes 0, both foci collapse onto the centre, and the ellipse is simply a circle.

Example Reading an ellipse centred at the origin

Analyse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ .

Centre

(0,\ 0)

, since nothing is subtracted from

x

y

The larger denominator, 25, sits under

x

, so

a^2 = 25 \Rightarrow a = 5

and the major axis is horizontal. Then

b^2 = 9 \Rightarrow b = 3

c^2 = a^2 - b^2 = 25 - 9 = 16

, so

c = 4

Vertices at

(\pm 5,\ 0)

; foci at

(\pm 4,\ 0)

⟹ e = c/a = 4/5 = 0.8

Example A shifted, vertical ellipse

Analyse $\dfrac{(x-2)^2}{64} + \dfrac{(y+1)^2}{100} = 1$ .

Centre

(2,\ -1)

— remember

(y+1)

means

k = -1

The larger denominator, 100, sits under

y

, so

a^2 = 100 \Rightarrow a = 10

and the major axis is vertical. Then

b^2 = 64 \Rightarrow b = 8

c^2 = 100 - 64 = 36

, so

c = 6

Because the major axis is vertical, the foci sit 6 units above and below the centre:

(2,\ 5)

and

(2,\ -7)

⟹ e = 6/10 = 0.6

This ellipse is noticeably rounder than the previous one, exactly as its smaller eccentricity predicts.

Example Building the equation from its parts

Find the equation of the ellipse centred at the origin with foci $(0,\ \pm 4)$ and a vertex at $(0,\ 5)$ .

Both foci lie on the

y

-axis, so the major axis is vertical and

a^2

belongs under

y

The vertex

(0,\ 5)

is 5 units from the centre, so

a = 5

The foci are 4 units out, so

c = 4

Rearranging

c^2 = a^2 - b^2

gives

b^2 = a^2 - c^2 = 25 - 16 = 9

⟹ x²/9 + y²/25 = 1

Note The reflection property, and where it is used

Anything emitted from one focus of an ellipse — light, sound, a shock wave — reflects off the curve and converges precisely on the other focus. This single property explains most of the ellipse's practical uses.

Medicine — a lithotripter places a kidney stone at one focus and the wave source at the other, so the energy concentrates on the stone.

Astronomy — planets, moons, and comets travel elliptical orbits with the central body at one focus.

Architecture — elliptical whispering galleries carry a whisper from one focus clearly to the other.

Note Mistakes to avoid

Assuming

a^2

is always under

x

— it is always the larger denominator, wherever it appears.

Using

c^2 = a^2 + b^2

; that belongs to the hyperbola. The ellipse uses a minus.

Placing the foci on the minor axis — they always lie on the major axis.

Misreading the centre:

(y+1)^2

means

k = -1

, not

+1

Summary

An ellipse is the set of points whose distances to two foci sum to a constant, equal to 2a.
In standard form, (h, k) is the centre and a² is always the larger denominator — the major axis follows whichever variable sits above it.
The focal distance comes from c² = a² − b², and the eccentricity e = c/a always lies strictly between 0 and 1.
When a = b the foci merge at the centre, e = 0, and the ellipse becomes a circle.