The Standard Equation of a Hyperbola

The Standard Equation of a Hyperbola — Moosa Academy

A hyperbola is the set of all points whose distances to two fixed foci differ by a constant amount. Unlike the ellipse, which closes into a single loop, the hyperbola breaks into two separate branches that spread apart forever, guided by a pair of straight lines called asymptotes.

Concept The defining property

For every point

P

on either branch,

|PF_1 - PF_2|

is the same constant — and that constant equals

2a

, the distance between the two vertices.

The centre is the midpoint between the foci, the vertices are the two points where the branches turn, and the transverse axis is the line through both — with the foci always lying beyond the vertices on it.

Concept The two standard forms

Horizontal — branches open left and right:

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

Vertical — branches open up and down:

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

Here is the crucial difference from the ellipse: orientation is decided by which term is positive, not by which denominator is larger. If the $x$ term is positive, the hyperbola opens horizontally; if the $y$ term is positive, it opens vertically. $a^2$ is always the denominator under the positive term, and $a$ may well be smaller than $b$ .

Note Every element, side by side

Feature	Horizontal	Vertical
Positive term	the x term	the y term
Vertices	(h ± a, k)	(h, k ± a)
Foci	(h ± c, k)	(h, k ± c)
Transverse axis	Horizontal, length 2a	Vertical, length 2a
Conjugate axis	Vertical, length 2b	Horizontal, length 2b
Asymptote slopes	± b/a	± a/b

In both cases the focal relationship is $c^2 = a^2 + b^2$ — a plus, the exact opposite of the ellipse's minus. This forces $c > a$ always, which is why the eccentricity $e = c/a$ of a hyperbola is always greater than 1.

Concept The asymptotes

The asymptotes are two straight lines through the centre that the branches approach ever more closely without ever touching. They are the single most useful tool for sketching a hyperbola.

Horizontal hyperbola:

y - k = \pm\dfrac{b}{a}(x - h)

Vertical hyperbola:

y - k = \pm\dfrac{a}{b}(x - h)

A reliable way to draw them: mark $a$ units along the transverse axis and $b$ units along the conjugate axis, draw the rectangle those four points define, and extend its diagonals. Those diagonals are the asymptotes, and the branches nestle into the corners.

Example A horizontal hyperbola

Analyse $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ .

Centre

(0,\ 0)

. The

x

term is positive, so the hyperbola opens horizontally.

a^2 = 16 \Rightarrow a = 4

and

b^2 = 9 \Rightarrow b = 3

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

, so

c = 5

Vertices

(\pm 4,\ 0)

; foci

(\pm 5,\ 0)

Asymptotes

y = \pm\dfrac{3}{4}x

⟹ e = c/a = 5/4 = 1.25

Example A shifted, vertical hyperbola

Analyse $\dfrac{(y+2)^2}{25} - \dfrac{(x-3)^2}{144} = 1$ .

Centre

(3,\ -2)

— the

(y+2)

means

k = -2

The

y

term is positive, so this one opens vertically. Note

a^2 = 25

is the smaller denominator here, which is perfectly normal for a hyperbola.

a = 5

and

b = 12

c^2 = 25 + 144 = 169

, so

c = 13

Vertices

(3,\ 3)

and

(3,\ -7)

; foci

(3,\ 11)

and

(3,\ -15)

Asymptotes

y + 2 = \pm\dfrac{5}{12}(x - 3)

⟹ e = 13/5 = 2.6

This hyperbola has a much larger eccentricity than the first, and its branches spread apart correspondingly more widely.

Example Building the equation from its parts

Find the equation of the hyperbola centred at the origin with vertices $(\pm 3,\ 0)$ and foci $(\pm 5,\ 0)$ .

Vertices and foci both lie on the

x

-axis, so the hyperbola is horizontal and the

x

term is positive.

a = 3

from the vertices, and

c = 5

from the foci.

Rearranging

c^2 = a^2 + b^2

gives

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

⟹ x²/9 − y²/16 = 1

Notice the rearrangement: for a hyperbola you subtract to find $b^2$ , whereas for an ellipse you would subtract to find $c^2$ . Keeping $c^2 = a^2 + b^2$ straight in your head prevents the most common error in these problems.

Note Where hyperbolas appear

Navigation and GPS — the difference in signal arrival times from two satellites places a receiver on a hyperbola; intersecting several of them pinpoints the position.

Astronomy — a comet moving fast enough to escape the Sun follows a hyperbolic path rather than a closed orbit.

Engineering — cooling towers are built as hyperboloids, giving strength with minimal material.

Physics — charged particles deflected by a nucleus trace hyperbolic trajectories.

Note Mistakes to avoid

Deciding orientation by which denominator is larger — for a hyperbola it is the sign that decides, not the size.

Using

c^2 = a^2 - b^2

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

Swapping the asymptote slopes: it is

\pm b/a

for horizontal,

\pm a/b

for vertical.

Forgetting that

e > 1

always — an answer below 1 signals an arithmetic slip.

Sketching a single closed curve; a hyperbola always has two separate branches.

Summary

A hyperbola is the set of points whose distances to two foci differ by the constant 2a, forming two separate branches.
The positive term decides the orientation: x positive opens horizontally, y positive opens vertically — and a² always sits under the positive term.
The focal relationship is c² = a² + b², so c > a and the eccentricity e = c/a is always greater than 1.
Asymptotes run through the centre with slopes ±b/a (horizontal) or ±a/b (vertical), and are the fastest route to an accurate sketch.