Impulse And Momentum

Two balls, having linear momenta \({\overrightarrow p _1} = p\widehat i\) and \({\overrightarrow p _2} = - p\widehat i\), undergo a collision in free space. There is no external force acting on the balls. Let \({\overrightarrow {p'} _1}\) and \({\overrightarrow {p'} _2}\) be their final momenta. The following option(s) is (are) NOT ALLOWED for any non-zero value of \(p,{a_1},{a_2},{b_1},{b_2},{c_1}\) and \({c_2}\) :