Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### A right triangle is shown in the graph.
![Graph Description]
The graph shows a right triangle with vertices at points \((a, b)\) and \((x, y)\). The horizontal leg of the triangle extends from point \((a, b)\) to \((x, b)\), measuring \(x - a\), and the vertical leg extends from point \((x, b)\) to \((x, y)\), measuring \(y - b\). The hypotenuse of the triangle, labeled \(c\), extends from \((a, b)\) to \((x, y)\).
### Part A:
Use the Pythagorean Theorem to derive the standard equation of the circle, with center at \((a, b)\) and a point on the circle at \((x, y)\). Show all necessary math work.
### Part B:
If \((a, b) = (5, -2)\) and \(c = 10\), determine the domain and range of the circle.
### Part C:
Is the point (10, 2) inside the border of the circle if \((a, b) = (5, -2)\) and \(c = 10\)? Explain using mathematical evidence.
---
#### Diagram Explanation:
- **Axes**: The horizontal and vertical axes are labeled as \(x\) and \(y\), respectively.
- **Triangle**: A right triangle with its right angle at \((x, b)\). The hypotenuse \(c\) connects \((a, b)\) and \((x, y)\).
- **Points**:
- \((a, b)\) is the center of the circle.
- \((x, y)\) is a point on the circumference of the circle.
- **Distances**:
- The horizontal leg has the length \(x - a\).
- The vertical leg has the length \(y - b\).
This detailed explanation and depiction will aid in understanding how to derive the circle's equation and solve subsequent parts conceptually and mathematically.](https://content.bartleby.com/qna-images/question/3e08c0ed-2478-4250-8ecc-80933702ca06/df9c009b-bedd-4e44-a8ab-f5ee13272dba/olospe_processed.png)
Transcribed Image Text:### A right triangle is shown in the graph.
![Graph Description]
The graph shows a right triangle with vertices at points \((a, b)\) and \((x, y)\). The horizontal leg of the triangle extends from point \((a, b)\) to \((x, b)\), measuring \(x - a\), and the vertical leg extends from point \((x, b)\) to \((x, y)\), measuring \(y - b\). The hypotenuse of the triangle, labeled \(c\), extends from \((a, b)\) to \((x, y)\).
### Part A:
Use the Pythagorean Theorem to derive the standard equation of the circle, with center at \((a, b)\) and a point on the circle at \((x, y)\). Show all necessary math work.
### Part B:
If \((a, b) = (5, -2)\) and \(c = 10\), determine the domain and range of the circle.
### Part C:
Is the point (10, 2) inside the border of the circle if \((a, b) = (5, -2)\) and \(c = 10\)? Explain using mathematical evidence.
---
#### Diagram Explanation:
- **Axes**: The horizontal and vertical axes are labeled as \(x\) and \(y\), respectively.
- **Triangle**: A right triangle with its right angle at \((x, b)\). The hypotenuse \(c\) connects \((a, b)\) and \((x, y)\).
- **Points**:
- \((a, b)\) is the center of the circle.
- \((x, y)\) is a point on the circumference of the circle.
- **Distances**:
- The horizontal leg has the length \(x - a\).
- The vertical leg has the length \(y - b\).
This detailed explanation and depiction will aid in understanding how to derive the circle's equation and solve subsequent parts conceptually and mathematically.
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