Exercise Questions

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This exercise contains 3 questions. Use the Questions tab to view and track them.

Key Concepts

This exercise focuses on the following concepts:

Hyperbola Fundamentals: Understanding the geometric definition and standard forms of hyperbola equations.
Key Components: Identifying the centre, vertices, co-vertices, and foci.

Transverse and Conjugate Axes: Determining the equations and lengths of the principal axes.

Eccentricity: Calculating the ratio $e = \frac{c}{a}$ and its relationship to the shape of the hyperbola.

Asymptotes: Finding the equations of the asymptotes from the standard form.
Equation Derivation: Constructing equations based on given foci, vertices, or asymptotes.
Calculus in Conics: Using differentiation to locate vertices or the centre of a conic section.

Important Formulas

Below are the key formulas used in this exercise:

Property	Horizontal Transverse Axis	Vertical Transverse Axis
Standard Equation	$\frac{( x - h ) ^{2}}{a ^{2}} - \frac{( y - k ) ^{2}}{b ^{2}} = 1$	$\frac{( y - k ) ^{2}}{a ^{2}} - \frac{( x - h ) ^{2}}{b ^{2}} = 1$

| Foci-Vertex Relation | $c^{2} = a^{2} + b^{2}$ | $c^{2} = a^{2} + b^{2}$ |

| Eccentricity ( $e$ ) | $e = \frac{c}{a} = \frac{a ^{2} + b ^{2}}{a} > 1$ | $e = \frac{c}{a} = \frac{a ^{2} + b ^{2}}{a} > 1$ |

| Vertices | $(h \pm a, k)$ | $(h, k \pm a)$ | | Foci | $(h \pm c, k)$ | $(h, k \pm c)$ | | Length of Transverse Axis | $2 a$ | $2 a$ | | Length of Conjugate Axis | $2 b$ | $2 b$ | | Asymptotes | $y - k = \pm \frac{b}{a} (x - h)$ | $y - k = \pm \frac{a}{b} (x - h)$ |

Worked Example

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

Identify: $a^{2} = 9 \Rightarrow a = 3$ , $b^{2} = 4 \Rightarrow b = 2$ , $c = 9 + 4 = 13$ .

Element	Value
Centre	$(0, 0)$
Vertices	$(\pm 3, 0)$
Foci	$(\pm 13, 0)$
Eccentricity	$e = \frac{13}{3}$
Asymptotes	$y = \pm \frac{2}{3} x$

Summary

This exercise provides a comprehensive review of hyperbolas. It transitions from basic identification of properties from standard equations to the more complex task of deriving equations from specific geometric constraints. A notable strategy included is the use of differentiation to find vertices, which is particularly useful for hyperbolas not centered at the origin or those expressed in general second-degree forms.

Key mastery involves understanding the relationship between the parameters $a$ , $b$ , and $c$ , and being able to write asymptote equations directly from the standard form.

Key Concepts

This exercise focuses on the following concepts:

Hyperbola Fundamentals: Understanding the geometric definition and standard forms of hyperbola equations.

Key Components: Identifying the centre, vertices, co-vertices, and foci.

Transverse and Conjugate Axes: Determining the equations and lengths of the principal axes.

Eccentricity: Calculating the ratio

e = \frac{c}{a}

and its relationship to the shape of the hyperbola.

Asymptotes: Finding the equations of the asymptotes from the standard form.

Equation Derivation: Constructing equations based on given foci, vertices, or asymptotes.

Calculus in Conics: Using differentiation to locate vertices or the centre of a conic section.

Important Formulas

Below are the key formulas used in this exercise:

Property	Horizontal Transverse Axis	Vertical Transverse Axis
Standard Equation	$\frac{( x - h ) ^{2}}{a ^{2}} - \frac{( y - k ) ^{2}}{b ^{2}} = 1$	$\frac{( y - k ) ^{2}}{a ^{2}} - \frac{( x - h ) ^{2}}{b ^{2}} = 1$

| Foci-Vertex Relation |

c^{2} = a^{2} + b^{2}

c^{2} = a^{2} + b^{2}

| Eccentricity ( $e$ ) |

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

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

| Vertices |

(h \pm a, k)

(h, k \pm a)

| | Foci |

(h \pm c, k)

(h, k \pm c)

| | Length of Transverse Axis |

2 a

2 a

| | Length of Conjugate Axis |

2 b

2 b

| | Asymptotes |

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

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

Element

Value

Centre

(0, 0)

Vertices

(\pm 3, 0)

Foci

(\pm 13, 0)

Eccentricity

e = \frac{13}{3}

Asymptotes

y = \pm \frac{2}{3} x

Summary

Key mastery involves understanding the relationship between the parameters

a

b

, and

c

, and being able to write asymptote equations directly from the standard form.

7.9 Exercise 7.8

Exercise Questions

Key Concepts

Important Formulas

Worked Example

Summary

7.9 Exercise 7.8

Exercise Questions

Key Concepts

Important Formulas

Worked Example

Summary