Chapter 16 Curvature-I

16.1 Exercise 9

16.1.1 Question 1

Find the radius of curvature at any point $(x,y)$ for the curve

1. $y^2=4ax$ (TU 2057)

2. $x^{2/3}+y^{2/3}=a^{2/3}$

3. $y=c\cos \text{h}\left(\frac{x}{c}\right)$

4. $xy=c^2$

5. $x=a\cos \theta ,y=a\sin \theta$

6. $x=a\cos \varphi ,y=b\sin \varphi$

16.1.2 Question 2

In the cycloid $x=a(\theta +\sin \theta ),y=a(1-\cos \theta )$ at $\theta =0$, prove that $\rho =4a$.

16.1.3 Question 3

Show that the radius of curvature at a point $(r,\theta )$ for the curve $r=a{e}^{\theta \cot \alpha }$ is $\rho =r\csc \alpha$.

16.1.4 Question 4

Find the radius of curvature of the curve $r=a(1-\cos \theta )$.

16.1.5 Question 5

Show that the radius of curvature for the curve $r^m=a^m\cos m\theta$ is $\frac{a^m}{(m+1)r^{m-1}}$.

16.1.6 Question 6

Find the radius of curvature at the point $(p,r)$ of the curve $r^{m+1}=a^{m}p$.

16.1.7 Question 7

Find the radius of curvature of the curve

1. $y^2=4x$ at the vertex $(0,0)$ (TU 2054)

2. $y^2=\frac{a+x}{a-x}.x^2$ at the origin

16.1.8 Question 8

Show that for the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the radius of curvature at the extremity of the major axis is equal to half the latus rectum. (TU 2055)