Chapter 17 Curvature-II

17.0.1 Question 9

Show that the radius of curvature of the curve $x^{2}y=a(x^2+\frac{a^2}{\sqrt{5}})$ is least for the point $x=a$ and its value there is $\frac{9a}{10}$.

17.0.2 Question 10

Find the radius of curvature of the curve $y=x^2(x-3)$ at the points where the tangent is parallel to $x$-axis.

17.0.3 Question 11

Find the radius of curvature of the following curves

1. $r^2=a^2{\cos }^2\theta$ at $\theta =0$

2. $r=a(\theta +\sin \theta )$ at $\theta =0$

17.0.4 Question 12

For any curve prove that

$$\begin{array}{cc}\frac{r}{\rho }& =\sin \varphi (1+\frac{d\varphi }{d\theta })\end{array}$$

17.0.5 Question 13

If ${\rho }_1$ and ${\rho }_2$ be the radii of curvature at the ends of a focal chord of the parabola $y^2=4ax$, prove that

$$\begin{array}{cc}{\rho }_1^{-2/3}+{\rho }_2^{-2/3}& =(2a{)}^{-2/3}\end{array}$$

17.0.6 Question 14

Find the chord of curvature through the pole for the following curves

1. $r=a(1+\cos \theta )$

2. $r=a{e}^{\theta \cot \alpha }$

3. $r^2=a^2\cos 2\theta$

17.0.7 Question 15

Show that the chord of curvature parallel to $y$-axis for the curve $y=c\cos \text{h}\frac{x}{c}$ is double of the ordinate.

17.0.8 Question 16

Find the coordinates of the centre of curvature and the evolute of the curves

1. $y^2=4ax$ (TU 2057, 2065)

2. $x=a{\cos }^3\theta ,y=a{\sin }^3\theta$

3. $xy=a^2$

4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ (TU 2061)

5. $x=a(\theta -\sin \theta ),y=a(1-\cos \theta )$

6. $x=a(\cos t+t\sin t),y=a(\sin t-t\cos t)$