Time and work. How many questions does Mini make in x minutes?

If in $x$ minutes, Mini makes $a$ questions, then, Mini makes one question in $\frac{x}{a}$ minutes.

In $x$ minutes vinay makes $a-y$ questions, then, Vinay makes a question in $\frac{x}{a-y}$minutes.

Then, if Mini makes a question in $\frac{x}{a}-2$ minutes and Vinay in $\frac{x}{a-y}-2$ minutes, it is given that in $x$ minutes Mini makes $2y$ questions more.

Basically we want to find $a$ in the question. This can be solved by taking: $$\frac{x}{\frac{x}{a}-2}-\frac{x}{\frac{x}{a-y}-2}=2y$$ Solving the required quadratic in $a$, gives us one of the correct options as: $$\boxed{a= \frac{2(x+y)-\sqrt{2x^2+4y^2}}{4}}$$