Question

unh Find the uns an vector, the homol vochor and the curvahure of the : a sin, 2:0 0 he pon) with parameter θ,

Answer 1

Equation of circle

$x^2+y^2=a^2$

Differentiate

$2x+2y\frac{dy}{dx}=0$

$\frac{dy}{dx}=-\frac{x}{y}$

$\left.\frac{dy}{dx}\right|_{\theta}=-\cot\theta$

A vector along the tangent direction would be

$\hat{T}\propto\hat{\imath}-\cot\theta\hat{\jmath}$

Make its modulus 1 by multiplying sin

$\hat{T}=\sin\theta\hat{\imath}-\cos\theta\hat{\jmath}$

Similarly, slope of normal is

$-\left.\left (\frac{dy}{dx} \right )^{-1}\right|_{\theta}=\tan\theta$

and hence

$\hat{N}=\cos\theta\hat{\imath}+\sin\theta\hat{\jmath}$

Curvature general formula

$\kappa=\frac{|\frac{d^2y}{dx^2}|}{\sqrt{1+\left ( \frac{dy}{dx} \right )^2}}$

For the given curve

$\frac{d^2y}{dx^2} =-\frac{y-xdy/dx}{y^2}=-\frac{y^2+x^2}{y^3}=-\frac{a^2}{y^3}$

$\kappa=\frac{\frac{a^2}{y^3}}{\left[1+\left ( \frac{x}{y} \right )^2\right]^{3/2}}=\frac{1}{a}$

as expected.