Let integers a , b [-3,3] be such that a + b 0 . Then the number of all possible ordered pairs (a, b), for which | z- a z+ b |=1 and | array ccc z+1 & & ^2 & z+ ^2 & 1 ^2 & 1 & z+ array |=1, z C , where and ^2 are the roots of x^2+x+1=0 , is equal to ________.

Step-by-step solution

aligned & a, b I,-3 a, b 3, a+b 0 & |z-a|=|z+b| & | array ccc z+1 & & ^2 & z+ ^2 & 1 ^2 & 1 & z+ array |=1 & | array ccc z & z & z & z+ ^2 & 1 ^2 & 1 & z+ array |=1 aligned aligned & z | array ccc 1 & 1 & 1 & z+ ^2 & 1 ^2 & 1 & z+ array |=1 & z | array ccc 1 & 0 & 0 & z+ ^2- & 1- ^2 & 1- ^2 & z+ - ^2 array |=1 & z^3=1 & z= , ^2, 1 aligned Now aligned & |1- a |=|1+ b | & 10 pairs aligned