Find a Cartesian equation for the curve given by the polar equation T = 4 sin 0. 73°F 1. (x - 2)² + y² + 4 = 0 2. x² + (y-2)² = 4 3. (x - 2)² + y² = 4 4. (x + 2)² + y² = 4 5. x² + (y + 2)² + 4 = 0 6. x² + (y-2)² + 4 = 0 7. (x + 2)² + y² + 4 = 0 8. x² + (y + 2)² = 4
Find a Cartesian equation for the curve given by the polar equation T = 4 sin 0. 73°F 1. (x - 2)² + y² + 4 = 0 2. x² + (y-2)² = 4 3. (x - 2)² + y² = 4 4. (x + 2)² + y² = 4 5. x² + (y + 2)² + 4 = 0 6. x² + (y-2)² + 4 = 0 7. (x + 2)² + y² + 4 = 0 8. x² + (y + 2)² = 4
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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![**Finding a Cartesian Equation for a Curve**
We are tasked with finding a Cartesian equation for the curve defined by the polar equation:
\[ r = 4 \sin \theta \]
**Multiple Choice Options:**
1. \((x - 2)^2 + y^2 + 4 = 0\)
2. \(x^2 + (y - 2)^2 = 4\)
3. \((x - 2)^2 + y^2 = 4\)
4. \((x + 2)^2 + y^2 = 4\)
5. \(x^2 + (y + 2)^2 + 4 = 0\)
6. \(x^2 + (y - 2)^2 + 4 = 0\)
7. \((x + 2)^2 + y^2 + 4 = 0\)
8. \(x^2 + (y + 2)^2 = 4\)
**Explanation for Solution:**
To find the Cartesian form, recall the polar to Cartesian coordinate transformations:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
From \( r = 4 \sin \theta \), we know that \( r = y \),
so substituting \( y = 4 \sin \theta \), we get \( y = 4 \cdot \frac{y}{r} \).
Simplifying for \( r \), we have:
\[ r = \frac{4y}{r} \]
Square both sides:
\[ r^2 = 4y \]
Using \( r^2 = x^2 + y^2 \), substitute to get:
\[ x^2 + y^2 = 4y \]
Moving terms yields:
\[ x^2 + (y^2 - 4y) = 0 \]
Complete the square:
\[ x^2 + (y - 2)^2 = 4 \]
Thus, the correct Cartesian equation is option 2:
\[ x^2 + (y - 2)^2 = 4 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4a40cee-b6e7-4175-94ff-7eded3f3eb47%2Fc1023bf1-1bdc-43d3-89b8-6c859806a372%2Fqcdoe63_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding a Cartesian Equation for a Curve**
We are tasked with finding a Cartesian equation for the curve defined by the polar equation:
\[ r = 4 \sin \theta \]
**Multiple Choice Options:**
1. \((x - 2)^2 + y^2 + 4 = 0\)
2. \(x^2 + (y - 2)^2 = 4\)
3. \((x - 2)^2 + y^2 = 4\)
4. \((x + 2)^2 + y^2 = 4\)
5. \(x^2 + (y + 2)^2 + 4 = 0\)
6. \(x^2 + (y - 2)^2 + 4 = 0\)
7. \((x + 2)^2 + y^2 + 4 = 0\)
8. \(x^2 + (y + 2)^2 = 4\)
**Explanation for Solution:**
To find the Cartesian form, recall the polar to Cartesian coordinate transformations:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
From \( r = 4 \sin \theta \), we know that \( r = y \),
so substituting \( y = 4 \sin \theta \), we get \( y = 4 \cdot \frac{y}{r} \).
Simplifying for \( r \), we have:
\[ r = \frac{4y}{r} \]
Square both sides:
\[ r^2 = 4y \]
Using \( r^2 = x^2 + y^2 \), substitute to get:
\[ x^2 + y^2 = 4y \]
Moving terms yields:
\[ x^2 + (y^2 - 4y) = 0 \]
Complete the square:
\[ x^2 + (y - 2)^2 = 4 \]
Thus, the correct Cartesian equation is option 2:
\[ x^2 + (y - 2)^2 = 4 \]
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