The greatest and the least value of |z1+z2|  if z1=24+7i  and |z2|=6  are respectively

Note that |z|  =  6  represents a circle. As |z2|  =  6,                    |z1+z2|=|z2−(−24−7i)|represent distance between a point on the circle |z|=6  and the point C(−24−7i).|z1+z2|  will be greatest and least at points B  and A  which are the end points of the diameter of the circle through C.  As OC=25,  CA=OC−OA=25−6=19  and CB=OC+OB=25+6=31.Alternative Solution:                    |z2|=6⇒z2=6eiθ  where θ εR.∴      |z1+z2|2=|24+7i+6(cos  θ+i  sin  θ)|2                           =(24+6  cos  θ)2+(7+6  sin  θ)2                           =576+36  cos2  θ+288  cos  θ+49                                                                    +36  sin2  θ+84  sin  θ                           =625+36+12(24  cos  θ  +7  sin  θ)                           =661+12(25)  sin  (θ+α)                                           [put 7=r  cos  α  and 24=r  sin  α ]                           =661+300  sin  (θ+α)Thus, greatest possible value of |z1+z2|2  is 661+300=961  and the least possible values if |z1+z2|  are 31 and 19.