Find the radius of curvature of the following curve at the given point. Then write the equation of the circle of curvature at the point. The radius of curvature at a point P is given by where x is the curvature at P. y= In 10x at x 10 The radius of curvature at xE - 1.01/101 10 (Type an exact answer, using radicals as needed) The equation of the circle of curvature at x= 10 (Type an exact answer, using radicals as needed.)
Find the radius of curvature of the following curve at the given point. Then write the equation of the circle of curvature at the point. The radius of curvature at a point P is given by where x is the curvature at P. y= In 10x at x 10 The radius of curvature at xE - 1.01/101 10 (Type an exact answer, using radicals as needed) The equation of the circle of curvature at x= 10 (Type an exact answer, using radicals as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The image contains a mathematical expression related to the circle of curvature. It begins with an instruction:
(Type an exact answer, using radicals as needed.)
Following this, the problem states:
The equation of the circle of curvature at \( x = \frac{1}{10} \) is
\[
(x - [ \, ])^2 + (y - [ \, ])^2 = [ \, ]
\]
(Type an exact answer, using radicals as needed.)
Explanation:
- The task involves determining the center and radius of the circle of curvature at the point \( x = \frac{1}{10} \) on a given curve.
- The equation provided is a general representation for a circle, \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
- The solution requires filling in the blanks with exact values, possibly involving radicals, to complete the equation for the circle of curvature.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca46498c-9039-4635-914d-0b10c3becf43%2F0bfe463f-7320-4e76-b46c-4dbdc1e7d706%2Fy8pt74f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression related to the circle of curvature. It begins with an instruction:
(Type an exact answer, using radicals as needed.)
Following this, the problem states:
The equation of the circle of curvature at \( x = \frac{1}{10} \) is
\[
(x - [ \, ])^2 + (y - [ \, ])^2 = [ \, ]
\]
(Type an exact answer, using radicals as needed.)
Explanation:
- The task involves determining the center and radius of the circle of curvature at the point \( x = \frac{1}{10} \) on a given curve.
- The equation provided is a general representation for a circle, \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
- The solution requires filling in the blanks with exact values, possibly involving radicals, to complete the equation for the circle of curvature.
![**Find the radius of curvature of the following curve at the given point. Then write the equation of the circle of curvature at the point. The radius of curvature at a point \( P \) is given by \(\frac{1}{\kappa}\), where \(\kappa\) is the curvature at \( P \).**
\[ y = \ln 10x \quad \text{at} \quad x = \frac{1}{10} \]
---
The radius of curvature at \( x = \frac{1}{10} \) is \(\frac{1}{|\kappa|} = 1.01\sqrt{101} \)
(Type an exact answer, using radicals as needed.)
The equation of the circle of curvature at \( x = \frac{1}{10} \) is
\[ (x - [ \, ])^2 + (y - ( \, ))^2 = ( \, )^2 \]
(Type an exact answer, using radicals as needed.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca46498c-9039-4635-914d-0b10c3becf43%2F0bfe463f-7320-4e76-b46c-4dbdc1e7d706%2Fyy4jkj8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the radius of curvature of the following curve at the given point. Then write the equation of the circle of curvature at the point. The radius of curvature at a point \( P \) is given by \(\frac{1}{\kappa}\), where \(\kappa\) is the curvature at \( P \).**
\[ y = \ln 10x \quad \text{at} \quad x = \frac{1}{10} \]
---
The radius of curvature at \( x = \frac{1}{10} \) is \(\frac{1}{|\kappa|} = 1.01\sqrt{101} \)
(Type an exact answer, using radicals as needed.)
The equation of the circle of curvature at \( x = \frac{1}{10} \) is
\[ (x - [ \, ])^2 + (y - ( \, ))^2 = ( \, )^2 \]
(Type an exact answer, using radicals as needed.)
Expert Solution
Step 1
Given:
at
Radius of curvature is given by .
To find:
a) Radius of curvature.
b) Circle of curvature.
Formula to find the radius of curvature:
Derivative Formula:
Step by step
Solved in 5 steps
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