If the normal to the curve $y = f(x)$ is parallel to the $x-$axis,then the correct statement is:

Show that the normal at any point $\theta$ to the curve $x=a \cos \theta+a \theta \sin \theta, y=a \sin \theta-a \theta \cos \theta$ is at a constant distance from the origin.

If the angle between the curves $y^2=4x$ and $y=e^{-x/2}$ is $\theta$,then $\operatorname{cosec}^2(\theta/2)=$

For $h, k \in N$, let $P(h, k)$ be the point of intersection of the curves $x^2 y - x^3 = 8$ and $y^3 - x y^2 = 32$. If $\theta$ is the acute angle between these two curves at $P$, then $\tan \theta =$

An equation of the tangent to the curve $y = x^4$ from the point $(2, 0)$ not on the curve is

What is the equation of the normal to the curve $y = x \log x$ which is parallel to the line $2x - 2y + 3 = 0$?