Hyperbola

A hyperbola is the set of points $P$ such that the absolute difference of distances from $P$ to two fixed foci is constant. The constant equals $2a$, the distance between the two vertices.

A hyperbola has two branches , one near each focus , that open in opposite directions. Its eccentricity is greater than $1$.

Standard form (horizontal transverse axis)

Place foci at $(\pm c, 0)$ and let $|\,|PF_1| - |PF_2|\,| = 2 a$ with $c > a > 0$. Setting $b^2 = c^2 - a^2$: $\boxed{\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.}$

Key features.

• Centre: origin.
• Vertices: $(\pm a, 0)$.
• Foci: $(\pm c, 0)$, $c = \sqrt{a^2 + b^2}$.
• Eccentricity: $e = c/a > 1$.
• Transverse axis: $x$-axis, length $2a$.
• Conjugate axis: $y$-axis, length $2b$.
• Asymptotes: $y = \pm \dfrac{b}{a} x$.
• Directrices: $x = \pm a/e$.
• Latus rectum: length $\dfrac{2 b^2}{a}$.

Standard form (vertical transverse axis)

If the transverse axis is along $y$: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1.$

Foci at $(0, \pm c)$, vertices $(0, \pm a)$, asymptotes $y = \pm \dfrac{a}{b} x$.

Asymptotes

A defining feature of a hyperbola is its pair of straight-line asymptotes: lines that the hyperbola approaches but never touches as $|x|, |y| \to \infty$. For $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, the asymptotes are $y = \pm \dfrac{b}{a} x$.

You get them by replacing the $1$ on the right side with $0$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 0 \Rightarrow y = \pm (b/a) x$.

Rectangular (equilateral) hyperbola

If $a = b$, the asymptotes are perpendicular ($y = \pm x$). Such a hyperbola is called rectangular or equilateral, eccentricity $\sqrt{2}$.

Worked examples

Example 1. Find vertices, foci, eccentricity, and asymptotes of $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$.

$a^2 = 9 \Rightarrow a = 3$. $b^2 = 16 \Rightarrow b = 4$. $c^2 = a^2 + b^2 = 25 \Rightarrow c = 5$. Vertices: $(\pm 3, 0)$. Foci: $(\pm 5, 0)$. $e = c/a = 5/3$. Asymptotes: $y = \pm (4/3) x$. Latus rectum: $2 b^2/a = 32/3$.

Example 2. Find vertices, foci of $\dfrac{y^2}{16} - \dfrac{x^2}{9} = 1$.

Vertical transverse axis. $a = 4, b = 3$. $c = 5$. Vertices: $(0, \pm 4)$. Foci: $(0, \pm 5)$. $e = 5/4$. Asymptotes: $y = \pm (4/3) x$.

Example 3. Equation of hyperbola with foci $(\pm 5, 0)$ and vertices $(\pm 3, 0)$.

$a = 3, c = 5 \Rightarrow b^2 = 25 - 9 = 16$. Equation: $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$.

Example 4. Equation of hyperbola with eccentricity $\sqrt{2}$ and vertices $(\pm 2, 0)$.

$a = 2, e = \sqrt{2} \Rightarrow c = 2 \sqrt{2} \Rightarrow b^2 = c^2 - a^2 = 8 - 4 = 4$. Equation: $\dfrac{x^2}{4} - \dfrac{y^2}{4} = 1$ (rectangular).

Example 5. Find the asymptotes of $9 x^2 - 16 y^2 = 144$.

Standardise: $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$. Asymptotes: $y = \pm (3/4) x$.

Try it yourself

1. Find vertices, foci, $e$ of $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$.
2. Find vertices, foci, $e$ of $\dfrac{y^2}{4} - \dfrac{x^2}{12} = 1$.
3. Asymptotes of $\dfrac{x^2}{36} - \dfrac{y^2}{49} = 1$.
4. Equation of hyperbola with foci $(\pm 4, 0)$ and $e = 2$.
5. Equation of hyperbola with vertices $(0, \pm 3)$ and foci $(0, \pm 5)$.
6. Length of latus rectum of $\dfrac{x^2}{25} - \dfrac{y^2}{144} = 1$.
7. Equation of hyperbola with foci $(\pm 5, 0)$ and the conjugate axis of length $8$.
8. Find the eccentricity of $4 x^2 - 5 y^2 = 20$.
9. Show that the eccentricity of a rectangular hyperbola is $\sqrt{2}$.
10. A hyperbola has asymptotes $y = \pm 2 x$ and passes through $(1, 0)$. Find its equation.
11. Find vertices, foci, $e$ of $9 y^2 - 4 x^2 = 36$.
12. The difference of focal distances of any point on the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is $\dots$.

Pitfalls / Tricks

• For hyperbola, $c^2 = a^2 + b^2$ (sum, not difference like ellipse).
• Eccentricity is always greater than $1$ , that is the signature of a hyperbola.
• Identify the transverse axis: the variable with the positive coefficient sets the axis.
• Insight. Hyperbolas and ellipses are algebraic siblings, differing only by a sign. Toggle the minus sign and you switch families.