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Suppose z=a+bi, where a and b are integers and i is the imaginary unit. We are given that |1+iz|=|1-iz| and |z-(13+15i)|<17. Find the largest possible value of a+b.

Suppose z=a+bi, where a and b are integers and i is the imaginary unit. We are given that |1+iz|=|1-iz| and |z-(13+15i)|<17. Find the largest possible value of a+b.

Grade:11

2 Answers

Akash Kumar Dutta
98 Points
11 years ago

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.

sonny singh
18 Points
11 years ago

|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).

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