If ( a + b ) = 15 and ( a - b ) = 3, then the value of ab 4 is?

Answer: Option C

$$\begin{array}{rl}& \left(\sqrt{a}+\sqrt{b}\right)=15\\ & \text{Square both sides}\\ & \Rightarrow a+b+2\sqrt{ab}=225\\ & \Rightarrow a+b=225-2\sqrt{ab}.....(i)\\ & \left(\sqrt{a}-\sqrt{b}\right)=3\\ & \text{Square both sides}\\ & \Rightarrow a+b-2\sqrt{ab}=9\\ & \Rightarrow a+b=9+2\sqrt{ab}.....(ii)\\ & \text{From (i) and (ii)}\\ & \Rightarrow 225-2\sqrt{ab}=9+2\sqrt{ab}\\ & \Rightarrow 216=4\sqrt{ab}\\ & \Rightarrow 54=\sqrt{ab}\\ & \text{Divided by 4 on both sides}\\ & \Rightarrow \frac{\sqrt{ab}}{4}=\frac{54}{4}\\ & \Rightarrow \frac{\sqrt{ab}}{4}=\frac{27}{2}\end{array}$$