Two particles $A$ and $B$ are projected from the same point with the same speed but at different angles $\alpha$ and $\beta$ with the horizontal, such that the maximum height of $A$ is two–third of the horizontal range of $B$. Then which of the following relations are true ?

$\frac{u^2{\sin }^2\alpha }{2g}=\frac{2}{3}{u}^2\frac{\sin 2\beta }{g} \Rightarrow {\sin }^2\alpha =\frac{4}{3}\sin 2\beta$

$0\le {\sin }^2\alpha \le 1 and \sin 2\beta =\frac{3}{4}{\sin }^2\alpha \Rightarrow \beta \le \frac{1}{2}{\sin }^{-1}\left(\frac{3}{4}\right)$                   

$3{\sin }^2\alpha =4\sin 2\beta$                                                               

$3(1-\cos 2\alpha )=8\sin 2\beta$                                                      

$$\begin{gathered} R_A=\frac{u^2}{g}\sin 2\alpha \\ So maximum value of R_A is \frac{u^2}{g} and this occurs when \alpha ={45}^0.Hence \beta =\frac{1}{2}{\sin }^{-1}\frac{3}{4}{\sin }^2{45}^0=\frac{1}{2}{\sin }^{-1}\frac{3}{8} \end{gathered}$$