SOLUTION: Let z=a+bi. Discuss the values of a and b so that the following is true: z(3-i) is a pure imaginary number.

Question 1155811: Let z=a+bi. Discuss the values of a and b so that the following is true: z(3-i) is a pure imaginary number.
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
.

It can be solved writing equations, but as an alternative, can be solved mentally, if you just know the properties of complex numbers.

    In order for z*(i) is a pure imaginary number, 

    it is necessary and sufficient that the number z be the complex conjugate to 3-i, multiplied by any pure imaginary number.



    In other words, the set of complex numbers, satisfying the given condition, 

    is   {z | z = i*r*(3+i)},  where "r" is any real number.


    Or, which is the same, the set of complex numbers z, satisfying the given condition 

    is the set of complex numbers of the form  z = r*(-1+3i), where "r" is any real number.



    In terms of z = a + bi,  a = -r,  b = 3r,  where "r" is any real number.

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On complex numbers, see introductory lessons
    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane

    - Solved problems on taking roots of complex numbers
    - Solved problems on arithmetic operations on complex numbers
    - Solved problem on taking square root of complex number
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Complex numbers".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

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