Let z ( z complex number) be one of the roots of the equation x^2- ( ₂ - ₂ ) x+ - =0 . If the harmonic mean of the roots is 2 and |z|=1 , find the sum of all values of in degrees when 0 < < 360^ .

Step-by-step solution

Let the roots be a-i b, a+i b , then a^2+b^2=1 (given .|z|=1 ) Also, aligned H.M. & =2 a^2+b^2 & =2 a aligned a= 1 2 Quadratic equation x^2-x(2 a)+a^2+b^2=0x^2-x+1=0 ...(1) x^2- ₂ ( )+ - =0 ...(2) On comparing equations (1) and (2), we get aligned ₂ ( ) & =1 and - =1 & =2 in ( - =1) 2 - & =1 2 ^2 + & =0 & = -1 2 or =0 & = 7 6 =210^ or 330^ or 180^ Sum & =720 aligned