Hyperbola

Learn hyperbolas with a sharp exam focus: definitions, standard forms, asymptotes, focal formulas, and intersection tricks.

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Asymptotes

Quick Facts

Key hyperbola facts you should recall instantly.

Definition

Absolute distance difference to foci is constant (2a)

Standard form

x²/a² − y²/b² = 1 or y²/a² − x²/b² = 1

Parameter relation

c² = a² + b², e = c/a (e>1)

Asymptotes

y = ±(b/a)x (horizontal) or y = ±(a/b)x (vertical)

Latus rectum

Length = 2b²/a

Hyperbola Calculator (Interactive)

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Core Knowledge

1) Definition & Standard Equations

From “difference of distances” to standard form.

Locus definition

A hyperbola is the set of points P such that the absolute difference of distances to two fixed points (foci) is constant:

|PF_1 - PF_2| = 2a

Here a is the semi-transverse axis length.

Standard equations

Horizontal transverse axis:

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Vertical transverse axis:

\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Parameter relation

c^2 = a^2 + b^2,\qquad e=\frac{c}{a}>1

Many mistakes happen when students accidentally use the ellipse formula c² = a² − b².

Hyperbola (horizontal): branches, vertices, foci

For x²/a² − y²/b² = 1, vertices are (±a,0) and foci are (±c,0), where c² = a² + b².

Conjugate hyperbola (same asymptotes)

The conjugate hyperbola shares the same pair of asymptotes. This often helps in symmetry/geometry arguments.

2) Asymptotes

Two dashed lines that control the “infinite” behavior.

Horizontal hyperbola

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\quad\Rightarrow\quad y=\pm\frac{b}{a}x

The slopes of the asymptotes are ±(b/a). If you know the asymptotes, you often recover the ratio b:a.

Vertical hyperbola

\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\quad\Rightarrow\quad y=\pm\frac{a}{b}x

Same idea, but the slope changes. Decide the orientation first.

Asymptotes & triangles

Many area/angle problems reduce to triangles formed by asymptotes and lines. Use slope-angle formulas and similarity.

Asymptotes (dashed): y = ±(b/a)x

As |x| grows, the hyperbola approaches its asymptotes—this is a key geometric handle in many exam questions.

3) Geometric Properties

Eccentricity, focal radius, axes, and latus rectum.

Eccentricity

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

Hyperbola eccentricity is always greater than 1. Larger e means the branches open more “narrowly”.

Focal radius (horizontal hyperbola)

For a point P(x₀,y₀) on the hyperbola, focal radius formulas often appear as:

r = |ex_0 \pm a|

The sign depends on which focus and which branch you use.

Axes

Transverse axis length: 2a
Conjugate axis length: 2b
Focal distance: 2c

Latus rectum

\text{Latus rectum length }=\frac{2b^2}{a}

For the hyperbola, it is a useful “fixed chord length” through a focus.

How to recognize types quickly

Positive term tells the transverse axis direction (x-term positive ⇒ horizontal).
Asymptote slopes help identify b/a or a/b.
Always keep the center at the origin in standard forms on this page.

Connection to lines

Hyperbola–line problems often use “eliminate then apply Δ/Vieta”. The asymptotes provide geometry intuition, but the scoring is usually algebraic.

4) Line–Hyperbola Intersections

Eliminate → quadratic → discriminant / Vieta.

Standard workflow

Write the line in a convenient form (often y = kx + m).
Substitute into the hyperbola and simplify to a quadratic in x.
Use Δ to classify secant/tangent/no-intersection.
Use Vieta for sums/products to compute invariants.

Tangency condition

Tangency occurs when the quadratic has a double root:

\Delta = 0

Asymptote-driven geometry

If a line is close to an asymptote, intersections can be “far away”. Use asymptotes to reason about signs and rough location, then confirm algebraically.

5) Common Pitfalls

Fix these and your hyperbola accuracy jumps immediately.

1) Mixing ellipse/hyperbola relations

Hyperbola: c² = a² + b², ellipse: c² = a² − b².

2) Wrong asymptote slope

For x²/a² − y²/b² = 1, the slope is b/a, not a/b.

3) Forgetting absolute value in the definition

The defining condition is |PF₁ − PF₂| = 2a. Without absolute value you can lose a full case.

Practice Quiz

Questions

Correct

Accuracy

A hyperbola is the set of points whose distances to two fixed points have a constant:

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For x²/a² − y²/b² = 1, the key parameter relation is:

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Which range is correct for hyperbola eccentricity e?

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For a horizontal hyperbola x²/a² − y²/b² = 1, the asymptotes are:

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For y²/a² − x²/b² = 1 (vertical hyperbola), the asymptotes are:

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The latus rectum length of a hyperbola is:

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Vertices of x²/a² − y²/b² = 1 are located at:

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When a line is tangent to a hyperbola after substitution, the quadratic has:

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Which statement is true about asymptotes?

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A conjugate hyperbola shares with the original hyperbola the same:

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