A right triangle is shown in the graph. Y (x, y) (a,b) b a a 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.
A right triangle is shown in the graph. Y (x, y) (a,b) b a a 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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 41E
Related questions
Question
![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e08c0ed-2478-4250-8ecc-80933702ca06%2Fdf9c009b-bedd-4e44-a8ab-f5ee13272dba%2Folospe_processed.png&w=3840&q=75)
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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