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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Question
Select all equations that represent a circle
![### Transcription of the Mathematical Equations
The following mathematical equations have been handwritten in the provided image. These equations can be used for analyzing and transforming functions, particularly those involving conic sections like ellipses and circles.
#### Equation 1 - Circle Equation:
\[
(x + 1)^2 + (y + 2)^2 = 5
\]
This equation represents a circle with its center at point \((-1, -2)\) and a radius \(\sqrt{5}\).
#### Equation 2 - General Ellipse Equation:
\[
\frac{x^2}{10} + \frac{y^2}{10} = 1
\]
This equation represents an ellipse centered at the origin with the semi-major and semi-minor axes both equal to \(\sqrt{10}\).
#### Equation 3 - Transformed / Simplified Form:
\[
x^2 + 2x - y^2 + 2y = 1
\]
This is a transformed version, representing a combination of terms after applying certain operations. Note that this is not a standard conic section form.
#### Equation 4 - Another Transformed Form:
\[
x^2 + 2x + y^2 + 6y = 1
\]
This is another transformed version, showcasing further manipulation or transformation of the initial variables.
### Explanation of the Equations
1. **Circle Equation**: The first equation is in the standard form of a circle in a Cartesian coordinate system. The general form of a circle equation is \[(x-h)^2 + (y-k)^2 = r^2\], where \((h, k)\) denotes the center of the circle and \(r\) is the radius. In this case, the center \((-1, -2)\) and the radius is \(\sqrt{5}\).
2. **Ellipse Equation**: The second equation is in the standard form of an ellipse. The general form for an ellipse centered at the origin (0,0) is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively. Since the denominators under both \(x^2\) and \(y^2\) are the same in this equation, this ellipse is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a5cc470-46ca-427c-a5a1-e87a305bfaad%2F745ff64e-145b-4224-a99a-66dc381c50e6%2F3wu12j.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription of the Mathematical Equations
The following mathematical equations have been handwritten in the provided image. These equations can be used for analyzing and transforming functions, particularly those involving conic sections like ellipses and circles.
#### Equation 1 - Circle Equation:
\[
(x + 1)^2 + (y + 2)^2 = 5
\]
This equation represents a circle with its center at point \((-1, -2)\) and a radius \(\sqrt{5}\).
#### Equation 2 - General Ellipse Equation:
\[
\frac{x^2}{10} + \frac{y^2}{10} = 1
\]
This equation represents an ellipse centered at the origin with the semi-major and semi-minor axes both equal to \(\sqrt{10}\).
#### Equation 3 - Transformed / Simplified Form:
\[
x^2 + 2x - y^2 + 2y = 1
\]
This is a transformed version, representing a combination of terms after applying certain operations. Note that this is not a standard conic section form.
#### Equation 4 - Another Transformed Form:
\[
x^2 + 2x + y^2 + 6y = 1
\]
This is another transformed version, showcasing further manipulation or transformation of the initial variables.
### Explanation of the Equations
1. **Circle Equation**: The first equation is in the standard form of a circle in a Cartesian coordinate system. The general form of a circle equation is \[(x-h)^2 + (y-k)^2 = r^2\], where \((h, k)\) denotes the center of the circle and \(r\) is the radius. In this case, the center \((-1, -2)\) and the radius is \(\sqrt{5}\).
2. **Ellipse Equation**: The second equation is in the standard form of an ellipse. The general form for an ellipse centered at the origin (0,0) is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively. Since the denominators under both \(x^2\) and \(y^2\) are the same in this equation, this ellipse is
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