Question

The perpendicular distance from origin to the normal at any point to the curve $x=a\left(\cos \theta +\theta \sin \theta \right)$. $y=a\left(\sin \theta -\theta \cos \theta \right)$

• A. $a$
• B. $a/2$
• C. $a/3$
• D. $2a$

Answer 1

The correct option is A $a$

$\frac{dy}{dx}=\frac{dy}{d\theta }\frac{d\theta }{dx}=tan\theta =slopeoftangent$

Slope of the normal $=-\cot \theta$ $=\tan (90+\theta )$

$[y-a(\sin \theta -\theta \cos \theta \left)\right]=\frac{-\cos \theta }{\sin \theta }(x-a(\cos \theta +a\sin \theta \left)\right)$

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

Now distance from $(0,0)$

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