Where i am going wrong in finding normal to curve?

Question

The question is

Find the perpendicular distance between the normal to the curve $$x=a\cos t+at\sin t, y=a\sin t-at\cos t$$ and the origin.

Equation is given in parameterized form.

My attempt

finding slope of tangent to the curve at point $\theta$ $$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}|t=\theta;\implies tant| t=\theta$$ therefore slope at $\theta$ is equal to $\tan\theta$ slope of normal :$$\implies m=-\cot \theta$$

finding equation of normal at point $(acos\theta+a\theta sin\theta, asin\theta-a\theta cos\theta)$ by using $y-y_1 =m(x-x_1)$

$$y-(asin\theta -a\theta cos\theta)=-cot\theta(x-(acos\theta+a\theta cos\theta))$$ on finding distance of this line from origin we get the answer $a$ but the answer in my book is $a/2$

can anyone please tell me, why i am going wrong?

Answer 1

The normal simplifies to become $$y\sin t+x\cos t-a=0$$ Therefore the distance is $a$ and not $\frac a2$, so you are right and the book is wrong.