Hyperbola: Asymptotes, Conjugate Axis

Hyperbola: Asymptotes, Conjugate Axis

How to find the equations of the asymptotes and the conjugate axis of the hyperbola: definition, formula, proof, examples, and their solutions.

Definition

The asymptotes of a hyperbola are the lines that the graph of the hyperbola follows, as x and y become large.

The asymptotes of a hyperbola are the lines
that the graph of the hyperbola follows,
as x and y become large.

Formula 1

If (x^2)/(a^2) - (y^2)/(b^2) = 1, then the conjugate axis is the height of the dashed box, which is made by the asymptotes and the vertices of the hyperbola.

If x2/a2 - y2/b2 = 1,
then the conjugate axis is the height of the dashed box
which is formed by the asymptotes
and the vertieces of the hyperbola.

(conjugate axis) = 2b.

Hyperbola: formula, foci, transverse axis

If the hyperbola is (x^2)/(a^2) - (y^2)/(b^2) = 1, then the equation of the asymptotes are y = +-(b/a)x.

If the hyperbola is x2/a2 - y2/b2 = 1,
then the equation of the asymptotes are
y = ±(b/a)x.

Example 1

The equation of a hyperbola is given below. (x^2)/9 - (y^2)/16 = 1 1. Find the lengths of the transverse and conjugate axes. 2. Find the equations of the asymptotes.

To find a, b, and c,
change the equation into the standard form.

x2/32 - y2/42 = 1.

Then a = 3 and b = 4.

y2 term is (-).

So (transverse axis) = 2⋅3 = 6.

Hyperbola: formula, foci, transverse axis

Then (conjugate axis) = 2⋅4 = 8.

So the asymptotes are y = ±(4/3)x.

You can see that
the graph of the hyperbola
follows its asymptotes y = ±(4/3)x.

And the transverse axis and the conjugate axis
perpendicularly bisect each other.

Formula 2

If (y^2)/(b^2) - (x^2)/(a^2) = 1, then the conjugate axis is the width of the dashed box, which is made by the asymptotes and the vertices of the hyperbola.

If y2/b2 - x2/a2 = 1,
then the conjugate axis is the width of the dashed box
which is formed by the asymptotes
and the vertieces of the hyperbola.

(conjugate axis) = 2a.

Hyperbola: formula, foci, transverse axis

If the hyperbola is (y^2)/(b^2) - (x^2)/(a^2) = 1, then the equation of the asymptotes are y = +-(b/a)x.

If the hyperbola is y2/b2 - x2/a2 = 1,
then the equation of the asymptotes are
y = ±(b/a)x.
[not y = ±(a/b)x]

Example 2

The equation of a hyperbola is given below. x^2 - y^2 = -4 1. Find the lengths of the transverse and conjugate axes. 2. Find the equations of the asymptotes.

To find a, b, and c,
change the equation into the standard form.

y2/22 - x2/22 = 1.

Then a = b = 2.

x2 term is (-).

So (transverse axis) = 2⋅2 = 4.
(2 from b)

Hyperbola: formula, foci, transverse axis

Then (conjugate axis) = 2⋅2 = 4.
(2 from a)

So the asymptotes are y = ±(2/2)x.

You can see that
the graph of the hyperbola
follows its asymptotes y = ±(2/2)x.

And the transverse axis and the conjugate axis
perpendicularly bisect each other.