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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.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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.
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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