The number of those tangents to the curve $$y^2-2 x^3-4 y+8=0$$ which pass through the point $$(1,2)$$ is

If the straight line $$x \cos \alpha+y \sin \alpha=p$$ touches the curve $$\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2$$ at the point $$(a, b)$$ on it and $$\frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}$$, then $$k=$$

Condition that 2 curves $$y^2=4 a x, x y=c^2$$ cut orthogonally is

A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is