Two variable complex numbers z ₁ & z ₂ with their arguments in (- , ] are such that | z ₁-(6+2 i ) | 2 2 and | z ₂-(4 2 + i 12 2 ) | k , where maximum value of ( z ₁ )= minimum value of (z₂ ) , then k is equal to

Step-by-step solution

Let the equation of OAB be y=m x | 6 ~m -2 1+ m ^2 |=2 2 (3 ~m -1)^2 (1+ m ^2 ) =2 9 m^2-6 m+1 1+m^2 =2 7 m^2-6 m-1=0 7 m^2-7 m+m-1=07 ~m ( ~m -1)+( m -1)=0 ~m =1 y=x must be tangent to the second circle | 12 2 -4 2 2 |=k k=8