Section 10.2

The Hyperbola

Hyperbolas round out the conic sections on Exam 4. If you mastered ellipses, most of the machinery is the same — the key twists are asymptotes, the plus sign in the foci formula, and remembering that a isn't necessarily the larger number. Get those differences down and you'll handle any conic the exam throws at you.

⚡ Quick Summary

If an ellipse is a squished circle, a hyperbola is more like a circle that got pulled apart into two separate curved pieces (called branches). Picture two mirrored boomerangs flying away from each other — that's the shape. While an ellipse cares about the *sum* of distances to two special points (foci), a hyperbola cares about the *difference*. Every point on a hyperbola has the same difference in distance to the two foci.

Standard Form (Horizontal Transverse Axis)

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

Center (h, k), vertices at (h ± a, k), branches open left and right

Standard Form (Vertical Transverse Axis)

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

Center (h, k), vertices at (h, k ± a), branches open up and down

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The equation looks a lot like an ellipse equation, but with a minus sign between the two fractions instead of a plus sign. That minus sign is your clue: "I'm a hyperbola!" The variable under the positive (first) fraction tells you which direction the branches open. Positive $x$ -term → opens left and right. Positive $y$ -term → opens up and down. And unlike ellipses, $a$ is simply whatever is under the positive term — it doesn't have to be the bigger number.

Hyperbolas come with asymptotes — invisible guide lines that the branches approach but never actually touch. Think of them like guardrails that the curves hug as they extend outward. To sketch a hyperbola, you draw a box centered at the middle of the hyperbola, then draw the diagonals of that box — those diagonals are the asymptotes. The branches curve away from the center and toward these lines.

To find the foci, use $c^2 = a^2 + b^2$ — notice it's plus (for ellipses it was minus). This means the foci are always farther from the center than the vertices. The eccentricity $e = c/a$ is always bigger than 1 for a hyperbola, which makes sense — the shape is "more open" than an ellipse.

Interactive Diagram

Key Formulas

Standard Form (Horizontal Transverse Axis)

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

Center (h, k), vertices at (h ± a, k), branches open left and right

Standard Form (Vertical Transverse Axis)

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

Center (h, k), vertices at (h, k ± a), branches open up and down

Asymptotes

\text{Horizontal: } y - k = \pm\frac{b}{a}(x - h) \\[4pt] \text{Vertical: } y - k = \pm\frac{a}{b}(x - h)

Slopes depend on which axis is the transverse axis — don't mix them up!

Foci Distance

c^2 = a^2 + b^2

PLUS, not minus — opposite of ellipses. Foci lie on the transverse axis, beyond the vertices.

Eccentricity

e = \frac{c}{a}, \quad e > 1

e near 1 → narrow branches; larger e → wider, more open branches

Key Takeaways

✓ $a^2$ is always under the positive (first) term — it's NOT necessarily the larger denominator
✓Foci formula: $c^2 = a^2 + b^2$ (plus for hyperbolas) — the opposite sign from ellipses
✓Asymptote slopes: $\pm b/a$ for horizontal, $\pm a/b$ for vertical — draw the reference rectangle to sketch quickly
✓Eccentricity $e = c/a > 1$ always; the positive term tells you which way the branches open

⚠️ Common Mistakes

✗Assuming the larger denominator is $a^2$ — that's an ellipse rule! For hyperbolas, $a^2$ is under the POSITIVE term, regardless of whether it's bigger or smaller
✗Using $c^2 = a^2 - b^2$ instead of $c^2 = a^2 + b^2$ — hyperbolas use plus, ellipses use minus
✗Swapping the asymptote slope formulas — horizontal hyperbolas use slopes $\pm b/a$ , vertical hyperbolas use slopes $\pm a/b$ ; mix them up and your sketch is wrong
✗Sign errors when completing the square with a negative leading coefficient — factoring $-9$ from $-9y^2 + 18y$ requires extra care with every sign
✗Getting $e < 1$ for a hyperbola — eccentricity of a hyperbola is always $e > 1$ ; if you get less than $1$ , recheck your $a$ and $c$ values

Next: Section 11.1 Builds on: 10.1

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10.1 The Ellipse

11.1 Sequences & Notation