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12. Look for Relationships Explain how to find the value of b in the equation of a hyperbola, given one focus at (0, 1) and one vertex at (0, 0.5).

12. Look for Relationships Explain how to find the value of b in the equation of a hyperbola, given one focus at (0, 1) and one vertex at (0, 0.5).

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 36E
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### Hyperbolas #### Definition: A hyperbola is the set of all points \(P\) such that the difference of the distances from any point \(P\) to two fixed points is constant. #### Graphs: **Horizontal Hyperbola:** - The graph shows a horizontal hyperbola with its transverse axis along the x-axis. - It has two vertices located at \((\pm a, 0)\). - Asymptotes are the lines that the hyperbola approaches but never meets. They are shown in blue and have the equations \(y = \pm \frac{b}{a}x\). - The foci, represented by \(F_1\) and \(F_2\), are located along the x-axis at \((\pm c, 0)\) where \(c = \sqrt{a^2 + b^2}\). **Vertical Hyperbola:** - This graph represents a vertical hyperbola with its transverse axis along the y-axis. - It has two vertices located at \((0, \pm a)\). - The asymptotes, shown in blue, have the equations \(y = \pm \frac{a}{b}x\). - The foci, represented by \(F_1\) and \(F_2\), are located along the y-axis at \((0, \pm c)\) where \(c = \sqrt{a^2 + b^2}\). #### Equation: **Horizontal:** \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] - Vertices: \((\pm a, 0)\) - Asymptotes: \(y = \pm \frac{b}{a}x\) - Foci: \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2}\) **Vertical:** \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] - Vertices: \((0, \pm a)\) - Asymptotes: \(y = \pm \frac{a}{b}x\) - Foci: \((0, \pm c)\), where \(c = \sqrt{a^2 + b^2}\) #### Do You Understand? 1
Transcribed Image Text:### Hyperbolas #### Definition: A hyperbola is the set of all points \(P\) such that the difference of the distances from any point \(P\) to two fixed points is constant. #### Graphs: **Horizontal Hyperbola:** - The graph shows a horizontal hyperbola with its transverse axis along the x-axis. - It has two vertices located at \((\pm a, 0)\). - Asymptotes are the lines that the hyperbola approaches but never meets. They are shown in blue and have the equations \(y = \pm \frac{b}{a}x\). - The foci, represented by \(F_1\) and \(F_2\), are located along the x-axis at \((\pm c, 0)\) where \(c = \sqrt{a^2 + b^2}\). **Vertical Hyperbola:** - This graph represents a vertical hyperbola with its transverse axis along the y-axis. - It has two vertices located at \((0, \pm a)\). - The asymptotes, shown in blue, have the equations \(y = \pm \frac{a}{b}x\). - The foci, represented by \(F_1\) and \(F_2\), are located along the y-axis at \((0, \pm c)\) where \(c = \sqrt{a^2 + b^2}\). #### Equation: **Horizontal:** \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] - Vertices: \((\pm a, 0)\) - Asymptotes: \(y = \pm \frac{b}{a}x\) - Foci: \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2}\) **Vertical:** \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] - Vertices: \((0, \pm a)\) - Asymptotes: \(y = \pm \frac{a}{b}x\) - Foci: \((0, \pm c)\), where \(c = \sqrt{a^2 + b^2}\) #### Do You Understand? 1
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