Understanding Hyperbolas: Concepts and Practice Problems

Chelsea

12/12/2025

Pre-Calculus

Hyperbolas

Subjects

Pre-Calculus

Ellipses & Hyperbola

Foci of a Hyperbola

•

Dec 12, 2025

•

4 pages

Understanding Hyperbolas: Concepts and Practice Problems

Chelsea

@zuncho

Hyperbolas are fascinating curves with unique properties that appear in... Show more

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Definition of a Hyperbola
A hyperbola is the set of all points (x, y) in a plane, the difference of whose
distances from two distinct

Definition and Standard Equation of a Hyperbola

A hyperbola consists of two separate curved parts called branches that open in opposite directions. It's defined as the set of all points where the difference of distances from two fixed points (the foci) is constant.

The standard equation of a hyperbola with center at (h, k) comes in two forms depending on orientation:

When the transverse axis is horizontal: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
When the transverse axis is vertical: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

The vertices are located a units from the center along the transverse axis, while the foci are c units from the center where $c^2 = a^2 + b^2$ . This relationship between a, b, and c is crucial for understanding hyperbola properties.

💡 When working with hyperbolas in general form $$Ax^2 + Cy^2 + Dx + Ey + F = 0$$ , you can quickly identify them because A and C will have different signs. This is your clue that you're dealing with a hyperbola!

Asymptotes and Graphing Hyperbolas

Every hyperbola has two asymptotes that intersect at the center and serve as guide lines for the curves. These asymptotes pass through the corners of an imaginary rectangle with dimensions 2a by 2b centered at (h, k).

For hyperbolas with a horizontal transverse axis, the asymptote equations are:

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

For hyperbolas with a vertical transverse axis:

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

To graph a hyperbola, first convert its equation to standard form and plot the center. Then mark the vertices, determine the asymptotes, and draw the curves approaching (but never touching) the asymptotes. The branches will always open along the transverse axis.

🔑 Remember this shortcut: the slopes of asymptotes are always ± $rise/run$ , where "run" corresponds to the denominator of the x-term in the standard equation. This makes graphing much faster!

Eccentricity and Additional Properties

The eccentricity of a hyperbola, calculated as $e = \frac{c}{a}$ , tells you about its shape. Since $c^2 = a^2 + b^2$ for hyperbolas, and c > a, the eccentricity is always greater than 1.

Hyperbolas with eccentricity values:

Close to 1: More pointed branches
Much larger than 1: Flatter, more open branches

The foci are always located along the transverse axis at a distance of c units from the center. This positioning creates the defining property of a hyperbola—the constant difference in distances from any point on the curve to the two foci.

👉 Unlike ellipses $where $c^2 = a^2 - b^2$$ , hyperbolas use the formula $c^2 = a^2 + b^2$ to relate these key measurements. This difference is essential for solving hyperbola problems!

Classifying Conics

You can quickly identify which conic section you're working with using the general form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ and these simple tests:

Circle: A = C (coefficients of x² and y² are equal)
Parabola: AC = 0 (either A or C equals zero, but not both)
Ellipse: AC > 0 (A and C have the same signs)
Hyperbola: AC < 0 (A and C have different signs)

Eccentricity also provides a neat way to classify all conic sections:

e = 0: Circle (perfect symmetry)
0 < e < 1: Ellipse (more circular when closer to 0)
e = 1: Parabola (the transition point)
e > 1: Hyperbola (more pointed when closer to 1)

🎯 When faced with a complex equation, don't rush to convert to standard form! First look at the coefficients of x² and y² to quickly classify the conic section—this can save you tons of work.

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Subjects

Pre-Calculus

Ellipses & Hyperbola

Foci of a Hyperbola

Pre-Calculus

•

Dec 12, 2025

•

4 pages

Understanding Hyperbolas: Concepts and Practice Problems

Chelsea

@zuncho

Hyperbolas are fascinating curves with unique properties that appear in various real-world applications from navigation to physics. They're defined by points whose difference in distances from two fixed points remains constant, creating distinctive curved shapes that open in opposite directions.

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Definition and Standard Equation of a Hyperbola

The standard equation of a hyperbola with center at (h, k) comes in two forms depending on orientation:

When the transverse axis is horizontal: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
When the transverse axis is vertical: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

💡 When working with hyperbolas in general form $$Ax^2 + Cy^2 + Dx + Ey + F = 0$$ , you can quickly identify them because A and C will have different signs. This is your clue that you're dealing with a hyperbola!

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Asymptotes and Graphing Hyperbolas

For hyperbolas with a horizontal transverse axis, the asymptote equations are:

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

For hyperbolas with a vertical transverse axis:

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

🔑 Remember this shortcut: the slopes of asymptotes are always ± $rise/run$ , where "run" corresponds to the denominator of the x-term in the standard equation. This makes graphing much faster!

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Eccentricity and Additional Properties

The eccentricity of a hyperbola, calculated as $e = \frac{c}{a}$ , tells you about its shape. Since $c^2 = a^2 + b^2$ for hyperbolas, and c > a, the eccentricity is always greater than 1.

Hyperbolas with eccentricity values:

Close to 1: More pointed branches
Much larger than 1: Flatter, more open branches

👉 Unlike ellipses $where $c^2 = a^2 - b^2$$ , hyperbolas use the formula $c^2 = a^2 + b^2$ to relate these key measurements. This difference is essential for solving hyperbola problems!

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Classifying Conics

You can quickly identify which conic section you're working with using the general form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ and these simple tests:

Circle: A = C (coefficients of x² and y² are equal)
Parabola: AC = 0 (either A or C equals zero, but not both)
Ellipse: AC > 0 (A and C have the same signs)
Hyperbola: AC < 0 (A and C have different signs)

Eccentricity also provides a neat way to classify all conic sections:

e = 0: Circle (perfect symmetry)
0 < e < 1: Ellipse (more circular when closer to 0)
e = 1: Parabola (the transition point)
e > 1: Hyperbola (more pointed when closer to 1)

🎯 When faced with a complex equation, don't rush to convert to standard form! First look at the coefficients of x² and y² to quickly classify the conic section—this can save you tons of work.