If the tangent drawn at the point $(\alpha, \beta)$ on the curve $x^{2/3} + y^{2/3} = 4$ is parallel to the line $\sqrt{3}x + y = 1$,then $\alpha^2 + \beta^2 =$

Let $M$ and $N$ be the number of points on the curve $y^{5}-9xy+2x=0$,where the tangents to the curve are parallel to the $x$-axis and $y$-axis,respectively. Then the value of $M + N$ equals $......$

Find the equation of all lines having slope $2$ and being tangent to the curve $y+\frac{2}{x-3}=0.$

At which point on the curve $y = x^2 + 3x$ should the tangent be drawn so that it passes through the point $(0, -9)$?

The tangent to the curve $y = x^{3} + 1$ at $(1, 2)$ makes an angle $\theta$ with the $y$-axis. Then the value of $\tan \theta$ is

If the tangent of the curve $y=e^{x}$ at a point $(c, e^{c})$ and the normal to the parabola $y^{2}=4x$ at the point $(1,2)$ intersect at the same point on the $x$-axis,then the value of $c$ is