Question

Find the angle at which the normal to the curve: $x=a(\cos \theta +\theta \sin \theta ),y=a(\sin \theta -\theta \cos \theta )$ at any point ${}^{\prime }{\theta }^{\prime }$ is incline to $x-axis$.

Answer 1

we have,

$\frac{dy}{dx}=\frac{dy}{d\theta }\cdot \frac{d\theta }{dx}=\tan \theta =$ slope of tangent

Therefore slope of normal to the curve $=-\cot \theta =\tan (90+\theta )$

Now equation of normal to the curve $[y-a(\sin \theta -\theta \cos \theta \left)\right]=-\frac{\cos \theta }{\sin \theta }(x-a(\cos \theta +a\sin \theta \left)\right)$

$\Rightarrow x\cos \theta +y\sin \theta =a\left(1\right)$

Now distance from $(0,0)$ to $x\cos \theta +y\sin \theta =a$ is

distance $\left(d\right)=\frac{0+0-a}{1}=a$