The horizontal range R of two projectiles are same when their maximum heights are H₁ and H₂. The value of R is (assume equal speed of projection)

Hint : For same range, sum of angle of projection is 90^o.

$H_1=\frac{u^2{\sin }^2\left({\theta }_1\right)}{2g}{H}_2=\frac{u^2{\sin }^2{\theta }_2}{2g}$

since R is same

$\begin{array}{l}\Rightarrow {\theta }_1+{\theta }_2={90}^{\circ }\\ \Rightarrow H_1=\frac{u^2{\sin }^2{\theta }_1}{2g},H_2=\frac{u^2{\cos }^2{\theta }_1}{2g}\end{array}$

Multiplying

$\begin{array}{l}{H}_1{H}_2=\frac{u^4{\sin }^2{\theta }_1{\cos }^2{\theta }_1}{4g}\\ \sqrt{H_1{H}_2}=\frac{u^2\sin {\theta }_1\cos {\theta }_1}{2g}\\ \sqrt{H_1{H}_2}=\frac{u^2\sin \left(2{\theta }_1\right)}{4g}\\ \Rightarrow R=4\sqrt{H_1{H}_2}\end{array}$