Training Conic Sections Hyperbolas
3 / 5

Hyperbolas

24 min Conic Sections

Hyperbolas

Hyperbola

The set of all points whose distances from two foci have a constant difference of $2a$.

Horizontal transverse axis: $$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

Vertical transverse axis: $$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$$

Key Relationships

$$c^2 = a^2 + b^2$$

Asymptotes (centered at origin, horizontal): $y = \pm \frac{b}{a}x$

Asymptotes (vertical): $y = \pm \frac{a}{b}x$

Example 1

Find the vertices, foci, and asymptotes: $\frac{x^2}{9}-\frac{y^2}{16}=1$.

$a=3$, $b=4$, $c = \sqrt{9+16} = 5$.

Vertices: $(\pm 3, 0)$. Foci: $(\pm 5, 0)$. Asymptotes: $y = \pm \frac{4}{3}x$.

Example 2

Write the equation: vertices $(0, \pm 4)$, foci $(0, \pm 5)$.

Vertical transverse. $a = 4$, $c = 5$, $b^2 = 25 - 16 = 9$.

$$\frac{y^2}{16} - \frac{x^2}{9} = 1$$

Example 3

Find the asymptotes: $\frac{y^2}{25} - \frac{x^2}{4} = 1$.

Vertical transverse: $y = \pm \frac{a}{b}x = \pm \frac{5}{2}x$.

Practice Problems

1. Vertices and foci: $\frac{x^2}{4}-\frac{y^2}{12}=1$.
2. Write the equation: vertices $(\pm 6,0)$, $b = 8$.
3. Find the asymptotes: $\frac{x^2}{49}-\frac{y^2}{9}=1$.
4. Is the transverse axis horizontal or vertical: $\frac{y^2}{16}-\frac{x^2}{9}=1$?
5. $c$ for $a=5$, $b=12$?
6. Write equation: foci $(\pm 10,0)$, vertices $(\pm 6,0)$.
7. Distinguish from ellipse: how does $c$ relate to $a$ and $b$?
8. Center and $a$ for $\frac{(x+1)^2}{25}-\frac{(y-3)^2}{4}=1$.
9. A hyperbola has asymptotes $y = \pm 2x$ and $a = 3$. Find $b$.
10. What is the eccentricity of a hyperbola?
11. Find the conjugate axis length: $\frac{x^2}{16}-\frac{y^2}{9}=1$.
12. Classify: $4x^2 - 9y^2 = 36$.
Show Answer Key

1. $a=2$, vertices $(\pm 2,0)$; $c=4$, foci $(\pm 4,0)$

2. $\frac{x^2}{36}-\frac{y^2}{64}=1$

3. $y = \pm \frac{3}{7}x$

4. Vertical

5. $c = \sqrt{25+144} = 13$

6. $b^2 = 100-36 = 64$; $\frac{x^2}{36}-\frac{y^2}{64}=1$

7. Hyperbola: $c^2 = a^2+b^2$; Ellipse: $c^2 = a^2-b^2$

8. Center $(-1,3)$, $a = 5$

9. $b/a = 2$, so $b = 6$

10. $e = c/a > 1$

11. $2b = 6$

12. Hyperbola (divide by 36: $\frac{x^2}{9}-\frac{y^2}{4}=1$)

Ellipses Classifying Conics & Applications