Hyperbola - Conic Sections and Properties

1️⃣ Understand the definition of a hyperbola and its components

2️⃣ Master hyperbola equations in standard and general forms

3️⃣ Identify key elements: center, vertices, foci, and asymptotes

4️⃣ Calculate eccentricity and understand c² = a² + b²

5️⃣ Apply hyperbola properties to real-world applications

1️⃣ Definition of a Hyperbola

A hyperbola is the locus of all points where the absolute difference of distances from two fixed points (foci) is constant. Unlike circles and ellipses, hyperbolas consist of two separate branches that extend infinitely.

$$|PF_1 - PF_2| = 2a$$ (constant) for any point P on the hyperbola

Key Difference from Ellipse: For hyperbolas, $$c^2 = a^2 + b^2$$ (addition), while for ellipses, $$c^2 = a^2 - b^2$$ (subtraction)

2️⃣ Hyperbola Equations and Types

a) Horizontal Hyperbola

Branches extend horizontally (left and right) - transverse axis is horizontal

Standard Form - Center at Origin

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

General Form - Center at (h,k)

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

Properties of Horizontal Hyperbola:

Vertices: $$(h \pm a, k)$$
Foci: $$(h \pm c, k)$$ where $$c^2 = a^2 + b^2$$
Transverse axis: horizontal with length $$2a$$
Conjugate axis: vertical with length $$2b$$
Asymptotes: $$y - k = \pm \frac{b}{a}(x - h)$$

b) Vertical Hyperbola

Branches extend vertically (up and down) - transverse axis is vertical

Standard Form - Center at Origin

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

General Form - Center at (h,k)

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

Properties of Vertical Hyperbola:

Vertices: $$(h, k \pm a)$$
Foci: $$(h, k \pm c)$$ where $$c^2 = a^2 + b^2$$
Transverse axis: vertical with length $$2a$$
Conjugate axis: horizontal with length $$2b$$
Asymptotes: $$y - k = \pm \frac{a}{b}(x - h)$$

Key Difference Between the Two Forms

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

Positive term under $$x^2$$
Branches extend left and right
Vertices on horizontal axis

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

Positive term under $$y^2$$
Branches extend up and down
Vertices on vertical axis

3️⃣ Geometric Relationships

Fundamental Relationship

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

$$a$$

Half-length of transverse axis

$$b$$

Half-length of conjugate axis

$$c$$

Distance from center to focus

4️⃣ Asymptotes

Asymptotes are straight lines that the hyperbola approaches as it extends to infinity. They help us sketch the hyperbola accurately.

Horizontal Hyperbola

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

Slopes: $$\pm \frac{b}{a}$$

Vertical Hyperbola

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

Slopes: $$\pm \frac{a}{b}$$

5️⃣ Eccentricity

Eccentricity Definition

$$e = \frac{c}{a} = \frac{\sqrt{a^2 + b^2}}{a}$$

Properties of Hyperbola Eccentricity:

$$e > 1$$ always (because $$c > a$$)
Higher $$e$$ means more "spread out" hyperbola
As $$e$$ approaches 1, the shape approaches a straight line
Large $$e$$ creates widely separated branches

6️⃣ Worked Examples

Example 1: Horizontal Hyperbola Analysis

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

Analysis:

• Center: $$(0, 0)$$
• $$a^2 = 9 \rightarrow a = 3$$
• $$b^2 = 16 \rightarrow b = 4$$
• $$c = \sqrt{a^2 + b^2} = \sqrt{9 + 16} = 5$$
• Vertices: $$(\pm 3, 0)$$
• Foci: $$(\pm 5, 0)$$
• Asymptotes: $$y = \pm \frac{4}{3}x$$
• Eccentricity: $$e = \frac{5}{3} \approx 1.67$$

Example 2: Vertical Hyperbola

Given: $$\frac{(y-1)^2}{9} - \frac{(x+2)^2}{4} = 1$$

Analysis:

• Center: $$(-2, 1)$$
• $$a^2 = 9 \rightarrow a = 3$$ (under y-term)
• $$b^2 = 4 \rightarrow b = 2$$
• $$c = \sqrt{9 + 4} = \sqrt{13}$$
• Vertices: $$(-2, 1 \pm 3) = (-2, 4)$$ and $$(-2, -2)$$
• Foci: $$(-2, 1 \pm \sqrt{13})$$
• Asymptotes: $$y - 1 = \pm \frac{3}{2}(x + 2)$$

7️⃣ Real-World Applications

Physics & Astronomy

Comet and asteroid trajectories
Particle paths in electric fields
Radio telescope antenna design
Sound wave propagation

Technology & Navigation

GPS and LORAN navigation
Radar and sonar systems
Architecture and structural design
Optics and lens systems

Key Takeaways

1️⃣ Hyperbolas have two branches with $$|PF_1 - PF_2| = 2a$$

2️⃣ Key relationship: $$c^2 = a^2 + b^2$$ (different from ellipse)

3️⃣ Horizontal: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$, Vertical: $$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$

4️⃣ Asymptotes guide the hyperbola's shape

5️⃣ Eccentricity $$e = \frac{c}{a} > 1$$ always

Hyperbola

All about Hyperbola with Simulation