Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, the tangent lines, and the normal lines. x² + y² = 25 (4,3), (3, 4) At (4,3)
Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, the tangent lines, and the normal lines. x² + y² = 25 (4,3), (3, 4) At (4,3)
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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### Interactive Geometry: Determining Tangents and Normals
At \((-3, 4)\)
Below are four graphs depicting a circle centered at the origin \((0,0)\) with different straight lines intersecting the circle. You are tasked with identifying which line represents the tangent to the circle at the specific point given.
1. **Top-Left Graph**
- The graph includes a circle centered at the origin with a line intersecting it. The y-axis ranges from \(10\) to \(-10\), and the x-axis ranges from \(10\) to \(-10\).
2. **Top-Right Graph**
- Similar to the top-left graph, this one depicts a circle centered at the origin with another line intersecting it. The axes here also span from \(-10\) to \(10\).
3. **Bottom-Left Graph**
- Again, a circle centered at the origin intersected by a different line. Axes limits are the same - ranging from \(-10\) to \(10\).
4. **Bottom-Right Graph**
- This graph displays a circle centered at the origin with yet another line intersecting it. The x and y axes extend from \(-10\) to \(10\) as well.
### Identify the Tangent Line
Select the graph which correctly shows the tangent line at the point \((-3, 4)\). A tangent line touches the circle at exactly one point.
### Input Fields
- **Tangent line:** \( y = \) [Enter the equation of the tangent line here]
- **Normal line:** \( y = \) [Enter the equation of the normal line here]
### Assistance
If you need help solving this problem, you can:
- **Read It:** Click the button to access detailed written instructions.
- **Watch It:** Click the button to watch a video tutorial on the topic.
---
**Note:** The correct answers must be determined based on the appropriate graph, which correctly shows the line tangential to the circle at the point \((-3, 4)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0711daf6-3b19-448b-99c0-3e7fe08f4fe3%2F16ad3690-cb62-4db2-9f5a-48f82c224656%2Fbcm98aj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Interactive Geometry: Determining Tangents and Normals
At \((-3, 4)\)
Below are four graphs depicting a circle centered at the origin \((0,0)\) with different straight lines intersecting the circle. You are tasked with identifying which line represents the tangent to the circle at the specific point given.
1. **Top-Left Graph**
- The graph includes a circle centered at the origin with a line intersecting it. The y-axis ranges from \(10\) to \(-10\), and the x-axis ranges from \(10\) to \(-10\).
2. **Top-Right Graph**
- Similar to the top-left graph, this one depicts a circle centered at the origin with another line intersecting it. The axes here also span from \(-10\) to \(10\).
3. **Bottom-Left Graph**
- Again, a circle centered at the origin intersected by a different line. Axes limits are the same - ranging from \(-10\) to \(10\).
4. **Bottom-Right Graph**
- This graph displays a circle centered at the origin with yet another line intersecting it. The x and y axes extend from \(-10\) to \(10\) as well.
### Identify the Tangent Line
Select the graph which correctly shows the tangent line at the point \((-3, 4)\). A tangent line touches the circle at exactly one point.
### Input Fields
- **Tangent line:** \( y = \) [Enter the equation of the tangent line here]
- **Normal line:** \( y = \) [Enter the equation of the normal line here]
### Assistance
If you need help solving this problem, you can:
- **Read It:** Click the button to access detailed written instructions.
- **Watch It:** Click the button to watch a video tutorial on the topic.
---
**Note:** The correct answers must be determined based on the appropriate graph, which correctly shows the line tangential to the circle at the point \((-3, 4)\).
![**Finding Equations for the Tangent and Normal Lines**
**Problem Statement:**
Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, the tangent lines, and the normal lines.
**Given Equation:**
\[ x^2 + y^2 = 25 \]
**Points Provided:**
- \( (4, 3) \)
- \( (3, 4) \)
- \((-4, 3)\)
- \((-3, 4)\)
- **At \( (4, 3) \)**
**Graphs and Diagrams:**
Four circles are shown with the center at the origin (0, 0) and radius = 5 (since \( 25 = 5^2 \)). Each circle has tangent and normal lines drawn at one of the identified points (i.e., (4, 3), (3, 4), (-4, 3), and (-3, 4)).
1. **Top Left Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (4, 3).
- The tangent line appears to be perpendicular to a radius at the point (4, 3).
2. **Top Right Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (3, 4).
- The tangent line appears to be perpendicular to a radius at the point (3, 4).
3. **Bottom Left Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (-4, 3).
- The tangent line appears to be perpendicular to a radius at the point (-4, 3).
4. **Bottom Right Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (-3, 4).
- The tangent line appears to be perpendicular to a radius at](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0711daf6-3b19-448b-99c0-3e7fe08f4fe3%2F16ad3690-cb62-4db2-9f5a-48f82c224656%2Fc74s92_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Equations for the Tangent and Normal Lines**
**Problem Statement:**
Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, the tangent lines, and the normal lines.
**Given Equation:**
\[ x^2 + y^2 = 25 \]
**Points Provided:**
- \( (4, 3) \)
- \( (3, 4) \)
- \((-4, 3)\)
- \((-3, 4)\)
- **At \( (4, 3) \)**
**Graphs and Diagrams:**
Four circles are shown with the center at the origin (0, 0) and radius = 5 (since \( 25 = 5^2 \)). Each circle has tangent and normal lines drawn at one of the identified points (i.e., (4, 3), (3, 4), (-4, 3), and (-3, 4)).
1. **Top Left Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (4, 3).
- The tangent line appears to be perpendicular to a radius at the point (4, 3).
2. **Top Right Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (3, 4).
- The tangent line appears to be perpendicular to a radius at the point (3, 4).
3. **Bottom Left Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (-4, 3).
- The tangent line appears to be perpendicular to a radius at the point (-4, 3).
4. **Bottom Right Graph:**
- Circle centered at the origin with given equation \( x^2 + y^2 = 25 \).
- Tangent and normal lines are drawn at the point (-3, 4).
- The tangent line appears to be perpendicular to a radius at
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