The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying

$\lfloor x\rfloor \cdot x=a\cdot x^2$

is $1050$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ and $x=x-\lfloor x\rfloor$ denotes the fractional part of $x$. What is $p+q$?