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26. A telescope is made with a hyperbolic mirror, shown graphed on the coordinate plane. To assemble the telescope correctly, the manufacturer needs to know the distance to the focus on the same side as the reflective side of the mirror. What is this distance? What equation represents the location of the mirror? SEE EXAMPLE 4 16 24 (12-50) (0-5) -16 -24

26. A telescope is made with a hyperbolic mirror, shown graphed on the coordinate plane. To assemble the telescope correctly, the manufacturer needs to know the distance to the focus on the same side as the reflective side of the mirror. What is this distance? What equation represents the location of the mirror? SEE EXAMPLE 4 16 24 (12-50) (0-5) -16 -24

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 29E
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## Example 4: Use a Hyperbola to Model a Real-World Situation ### Hyperbolic Mirror in Telescopes Some telescopes use a hyperbolic mirror. In these systems, light is directed inward at the hyperbolic mirror and reflected through a hole in the parabolic mirror towards a focus, where it can be captured by a detector. Below is an illustration of this setup. **Diagram Explanation:** 1. **Parabolic Primary Mirror**: Shown on the left side reflecting incoming light. 2. **Hyperbolic Secondary Mirror**: Positioned in such a way to reflect the focused light towards the detector. 3. **Focal Points**: Indicated as points `F` and another focus point. ### Analytical Method We start by placing the hyperbolic mirror on a coordinate plane. We aim to find the equation representing the mirror and the distance of the detector from the mirror. **Steps and Equations:** 1. Identify vertices and given points on the hyperbola. 2. One of the vertices is at (5, 0), hence `a = 5`. A given point on the hyperbola is (6, 3). **Equation Derivation:** \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Substituting \( (6, 3) \) into the hyperbole equation: \[ \frac{6^2}{5^2} - \frac{3^2}{b^2} = 1 \] \[ \frac{36}{25} - \frac{9}{b^2} = 1 \] Solving for `b^2`: \[ \frac{36}{25} - 1 = \frac{9}{b^2} \] \[ \frac{36 - 25}{25} = \frac{9}{b^2} \] \[ \frac{11}{25} = \frac{9}{b^2} \] \[ 11b^2 = 225 \] \[ b^2 = 20.5 \] Thus, the equation of the hyperbolic mirror is: \[ \frac{x^2}{25} - \frac{y^2}{20.5} = 1 \] ### Determining the Detector's Position The detector must be placed at the focus of the
Transcribed Image Text:## Example 4: Use a Hyperbola to Model a Real-World Situation ### Hyperbolic Mirror in Telescopes Some telescopes use a hyperbolic mirror. In these systems, light is directed inward at the hyperbolic mirror and reflected through a hole in the parabolic mirror towards a focus, where it can be captured by a detector. Below is an illustration of this setup. **Diagram Explanation:** 1. **Parabolic Primary Mirror**: Shown on the left side reflecting incoming light. 2. **Hyperbolic Secondary Mirror**: Positioned in such a way to reflect the focused light towards the detector. 3. **Focal Points**: Indicated as points `F` and another focus point. ### Analytical Method We start by placing the hyperbolic mirror on a coordinate plane. We aim to find the equation representing the mirror and the distance of the detector from the mirror. **Steps and Equations:** 1. Identify vertices and given points on the hyperbola. 2. One of the vertices is at (5, 0), hence `a = 5`. A given point on the hyperbola is (6, 3). **Equation Derivation:** \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Substituting \( (6, 3) \) into the hyperbole equation: \[ \frac{6^2}{5^2} - \frac{3^2}{b^2} = 1 \] \[ \frac{36}{25} - \frac{9}{b^2} = 1 \] Solving for `b^2`: \[ \frac{36}{25} - 1 = \frac{9}{b^2} \] \[ \frac{36 - 25}{25} = \frac{9}{b^2} \] \[ \frac{11}{25} = \frac{9}{b^2} \] \[ 11b^2 = 225 \] \[ b^2 = 20.5 \] Thus, the equation of the hyperbolic mirror is: \[ \frac{x^2}{25} - \frac{y^2}{20.5} = 1 \] ### Determining the Detector's Position The detector must be placed at the focus of the
### Telescope Hyperbolic Mirror Analysis #### Problem Statement A telescope is made with a hyperbolic mirror, as depicted on the coordinate plane below. To assemble the telescope correctly, the manufacturer needs to know the distance to the focus on the same side as the reflective side of the mirror. The key questions the manufacturer must answer are: 1. What is this distance? 2. What equation represents the location of the mirror? #### Graph and Hyperbolic Equation The provided coordinate plane features a hyperbola graphed with the following key points: - The graph spans from \( y = 0 \) to \( y = -24 \) on the y-axis in steps of -8. - The x-axis features points starting from \( x = -8 \) to \( x = 24 \), transitioning between -8 and 24 in steps of 8. - The mirror's graph intersects the y-axis at \( y = -5 \) and includes a notable point at approximately \( (12, -5) \). - Coordinates are specified at: - \( (0, -5) \) - \( (8, -16) \) - \( (-12, -50) \) The hyperbolic mirror's equation often takes the standard form: \[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = -1 \] Where \((h, k)\) is the center, and \(a\) and \(b\) are constants related to the distances between vertices and foci. #### Questions and Interpretations 1. **Distance Calculation**: Evaluate the distance from the reflective side of the mirror to the focus, using the points given from the graph. 2. **Equation Representation**: Deduce the precise hyperbolic equation which best fits the graphical representation provided. To find exact values of the distance and the equation, consult Example 4 in your provided resource materials and apply relevant formulas and functions accordingly. #### Conclusion Understanding the reflective properties and the precise location is crucial for the telescope's functionality. An accurate calculation ensures optimal assembly and proper focus for clear observations. For further details and step-by-step problem-solving methods, refer to Example 4 in your text.
Transcribed Image Text:### Telescope Hyperbolic Mirror Analysis #### Problem Statement A telescope is made with a hyperbolic mirror, as depicted on the coordinate plane below. To assemble the telescope correctly, the manufacturer needs to know the distance to the focus on the same side as the reflective side of the mirror. The key questions the manufacturer must answer are: 1. What is this distance? 2. What equation represents the location of the mirror? #### Graph and Hyperbolic Equation The provided coordinate plane features a hyperbola graphed with the following key points: - The graph spans from \( y = 0 \) to \( y = -24 \) on the y-axis in steps of -8. - The x-axis features points starting from \( x = -8 \) to \( x = 24 \), transitioning between -8 and 24 in steps of 8. - The mirror's graph intersects the y-axis at \( y = -5 \) and includes a notable point at approximately \( (12, -5) \). - Coordinates are specified at: - \( (0, -5) \) - \( (8, -16) \) - \( (-12, -50) \) The hyperbolic mirror's equation often takes the standard form: \[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = -1 \] Where \((h, k)\) is the center, and \(a\) and \(b\) are constants related to the distances between vertices and foci. #### Questions and Interpretations 1. **Distance Calculation**: Evaluate the distance from the reflective side of the mirror to the focus, using the points given from the graph. 2. **Equation Representation**: Deduce the precise hyperbolic equation which best fits the graphical representation provided. To find exact values of the distance and the equation, consult Example 4 in your provided resource materials and apply relevant formulas and functions accordingly. #### Conclusion Understanding the reflective properties and the precise location is crucial for the telescope's functionality. An accurate calculation ensures optimal assembly and proper focus for clear observations. For further details and step-by-step problem-solving methods, refer to Example 4 in your text.
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