Complex Help (1 Viewer)

[Green Yoda]

http://imgur.com/a/dlfyL

Help with all parts

 

[calamebe]

For i), note that z3-z2 is vector BC, and z3 is vector OC. As the radius is perpendicular to the tangent, angle OCB will be pi/2. Thus, (z3-z2)/z3 is imaginary, and similarly so is (z2-z1)/z2.
ii) This is just tangents from a point have equal lengths.
iii) Here, I would prove that OB and AC are perpendicular. This can be done by letting the point of intersection of AC and OB be D, and then proving that ADB and CDB are congruent. This can be done by proving that OAB and OCB are congruent, then that OAB is similar to ADB and CDB. This will allow you to prove that OB is perpendicular to AC.
iv) You know that OB is the diameter, as it subtends a right angle at OAB and ACB. So the centre will be (z2)/2.

 

[Green Yoda]

For i), note that z3-z2 is vector BC, and z3 is vector OC. As the radius is perpendicular to the tangent, angle OCB will be pi/2. Thus, (z3-z2)/z3 is imaginary, and similarly so is (z2-z1)/z2.
ii) This is just tangents from a point have equal lengths.
iii) Here, I would prove that OB and AC are perpendicular. This can be done by letting the point of intersection of AC and OB be D, and then proving that ADB and CDB are congruent. This can be done by proving that OAB and OCB are congruent, then that OAB is similar to ADB and CDB. This will allow you to prove that OB is perpendicular to AC.
iv) You know that OB is the diameter, as it subtends a right angle at OAB and ACB. So the centre will be (z2)/2.

Can you explain this part?

 

[calamebe]

Can you explain this part?

Yeah sure. So all we need in this question is the argument. arg((z3-z2)/z3)=arg(z3-z2)-arg(z3), or arg(BC)-arg(OC). Now, as vector BC is perpendicular to vector OC, this means that if we translate them both to the origin, there will be an angle of ± pi/2 between them. In other words, arg(BC)-arg(OC) = ± pi/2. So that means that (z3-z2)/z3 will be imaginary.

 

[He-Mann]

Can you explain this part?

The argument of a complex number is the angle starting from the positive real axis and going anti-clockwise (if positive) or clockwise (if negative). Clearly, an argument of +- pi/2 means the complex number is purely imaginary.