06_Ex 6.6

Exercise Questions

Questions	Links
Q1. Find an equation of the hyperbola with the given…	6.6 Q-1
Q2. Find the centre, foci, eccentricity, vertices…	6.6 Q-2
Q3. Let $0<\mathbf{a}<\mathbf{c}, \mathbf{F}(-\mathbf{c}, 0)$ and $\mathbf{F}^{\prime}(\mathbf{c}, 0)$ be two fixed points…	6.6 Q-3
Q4. Using Question 3 . Find an equation of hyperbola…	6.6 Q-4
Q5. For any point on the hyperbola the difference of…	6.6 Q-5
Q6. Two listening posts hear the sound of an enemy gun…	6.6 Q-6

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Overview

This exercise focuses on understanding and solving problems related to hyperbolas, a fundamental topic in conic sections. Hyperbolas are used in various scientific and real-life applications, such as satellite positioning, astronomy, and acoustics. Through these questions, we explore properties like vertices, foci, directrices, and their equations.

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Key Concepts

1. Definition of Hyperbola: A hyperbola is the locus of points such that the absolute difference of distances from two fixed points (foci) is constant.

2. Standard Equations:

   • Horizontal Hyperbola:

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

• Vertical Hyperbola:

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

3. Components:

   • Vertices: Points where the hyperbola intersects its transverse axis.
   • Foci: Fixed points, each located at a distance $c$ from the center.
   • Directrices: Lines associated with the eccentricity ($e$) of the hyperbola.

4. Relationships:

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

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Important Formulas

1. Eccentricity:

$$e = \frac{c}{a}, \quad \text{where } c = \sqrt{a^2 + b^2}.$$

2. Vertices and Foci:
   • Horizontal Hyperbola:

$$\text{Vertices: } (\pm a, 0), \quad \text{Foci: } (\pm c, 0).$$

• Vertical Hyperbola:

$$\text{Vertices: } (0, \pm a), \quad \text{Foci: } (0, \pm c).$$

3. Directrices:
   • Horizontal Hyperbola:

$$x = \pm \frac{a}{e}.$$

• Vertical Hyperbola:

$$y = \pm \frac{a}{e}.$$

4. General Form: Transformations can adjust the center of the hyperbola from $(0, 0)$ to $(h, k)$, resulting in:

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

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Tips and Tricks

1. Identifying the Axis:

   • The term with the positive denominator in the standard equation determines the transverse axis (e.g., $\frac{x^2}{a^2}$ indicates a horizontal axis).

2. Symmetry: Hyperbolas are symmetric with respect to both their axes and the center.

3. Directrix-Eccentricity Relation: Use the eccentricity ($e$) to find the directrix, which is perpendicular to the transverse axis.

4. Graphing: Sketching the hyperbola involves plotting vertices, foci, and asymptotes for guidance.

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Summary

• A hyperbola is defined as the set of points where the difference of distances to two foci is constant.
• The transverse axis determines the orientation of the hyperbola (horizontal or vertical).
• The relationship $c^2 = a^2 + b^2$ connects the semi-major axis ($a$), semi-minor axis ($b$), and the distance to the focus ($c$).
• Key components include vertices, foci, directrices, and eccentricity.
• Equations can be transformed by changing the center to any arbitrary point $(h, k)$.

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Reference

By Sir Shahzad Sair:

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