Dear Vivek,

From the first condition we get b=0 by putting z=a+ib....
hence z lies in the x axis.
hence for a+b to be max...a must be max.
also from second eq. is a circle and for a to be max...
the circle eq needs to be
=> (x-13)^2  +  (y-15)^2 = 17^2
=> putting y=0..since b=0
=> x - 13= (+-)8
hence x=21,5
For a+b to be max x=21
Hence max value= 21+0=21(ANS)

Regards.

|a+b|=|a+ib-ib+b-13-15i+13+15i|

=|(a+ib -13 -15i) + (b-ib) + (13+15i)| 

<=| (a+ib) - 13-15i|+|b-ib|+|13+15i|

= 17 +0 + root(394) = ans.= 17+root(394).