Chapter 17 Curvature-II

17.0.1 Question 9

Show that the radius of curvature of the curve \(x^2y=a\left(x^2 + \frac{a^2}{\sqrt{5}}\right)\) 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

\[\begin{equation*} \begin{split} \frac{r}{\rho} &= \sin \phi \left(1 + \frac{d\phi}{d\theta}\right) \end{split} \end{equation*}\]

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

\[\begin{equation*} \begin{split} \rho_1^{-2/3} + \rho_2^{-2/3} &= (2a)^{-2/3} \end{split} \end{equation*}\]

17.0.6 Question 14

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

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

  2. \(r=ae^{\theta \cot \alpha}\)

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