The equation of the normal drawn to the curve $y^3=4 x^5$ at the point $(4,16)$ is

The lines tangent to the curves $y^3 - x^2y + 5y - 2x = 0$ and $x^4 - x^3y^2 + 5x + 2y = 0$ at the origin intersect at an angle $\theta$ equal to

Find the points at which the tangent to the curve $y=x^{3}-3x^{2}-9x+7$ is parallel to the $x$-axis.

Find the coordinates of the point on the curve $\sqrt{x}+\sqrt{y}=4$ at which the tangent is equally inclined to the axes.

If the curves $y = x^3 - 3x^2 - 8x - 4$ and $y = 3x^2 + 7x + 4$ touch each other at a point $P$,then the equation of the common tangent at $P$ 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.