Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Problem Statement:**
Find an equation of the hyperbola having foci at \((2, -2)\) and \((10, -2)\) and vertices at \((5, -2)\) and \((7, -2)\).
**Description:**
- The problem involves finding the equation of a hyperbola based on given foci and vertices.
- The coordinates of the foci are \((2, -2)\) and \((10, -2)\).
- The coordinates of the vertices are \((5, -2)\) and \((7, -2)\).
**Diagram Explanation:**
- The figure shown consists of a rectangular input box where the solution or equation is to be entered.
- To the right of the input box, there are options with visual symbols, which likely indicate formatting or input choices for the equation. The meaning of each symbol is not specified.
**Mathematical Approach:**
1. **Calculate the Center:**
The center \((h, k)\) of the hyperbola is the midpoint of the vertices \((5, -2)\) and \((7, -2)\).
2. **Distance Calculations:**
- The distance between the vertices (transverse axis) gives the value of \(2a\).
- The distance between the foci gives the value of \(2c\).
3. **Standard Form of Equation:**
The equation of a hyperbola centered at \((h, k)\) with a horizontal transverse axis is given by:
\[
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
\]
4. **Solve for \(b^2\):**
\(c^2 = a^2 + b^2\) to find \(b^2\).
5. **Substitute Values:**
Substitute the values of \(h, k, a^2,\) and \(b^2\) into the standard form to get the equation of the hyperbola.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2039606c-ddf8-4dfc-99c4-5a13e28c47b4%2F9a8eb799-b24e-45d6-8756-3d928fb109fb%2Fgboer5b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find an equation of the hyperbola having foci at \((2, -2)\) and \((10, -2)\) and vertices at \((5, -2)\) and \((7, -2)\).
**Description:**
- The problem involves finding the equation of a hyperbola based on given foci and vertices.
- The coordinates of the foci are \((2, -2)\) and \((10, -2)\).
- The coordinates of the vertices are \((5, -2)\) and \((7, -2)\).
**Diagram Explanation:**
- The figure shown consists of a rectangular input box where the solution or equation is to be entered.
- To the right of the input box, there are options with visual symbols, which likely indicate formatting or input choices for the equation. The meaning of each symbol is not specified.
**Mathematical Approach:**
1. **Calculate the Center:**
The center \((h, k)\) of the hyperbola is the midpoint of the vertices \((5, -2)\) and \((7, -2)\).
2. **Distance Calculations:**
- The distance between the vertices (transverse axis) gives the value of \(2a\).
- The distance between the foci gives the value of \(2c\).
3. **Standard Form of Equation:**
The equation of a hyperbola centered at \((h, k)\) with a horizontal transverse axis is given by:
\[
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
\]
4. **Solve for \(b^2\):**
\(c^2 = a^2 + b^2\) to find \(b^2\).
5. **Substitute Values:**
Substitute the values of \(h, k, a^2,\) and \(b^2\) into the standard form to get the equation of the hyperbola.
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