# Thread: Complex Number question

1. ## Complex Number question

Hi everyone, anyone fancy helping out on yet another question

A complex number $z$ is represented by the point P on an Argand diangram, $|z-5-12i| = 3$.

a. Sketch the locus of P.

This I could do, circle, centre (5, 12), radius of 3.

b. Find the maximum and minimum values for $|z|$.

For this I found the distance of the centre of the circle from the origin, and then $\pm$ the radius to get 10 and 16, is this correct?

c. Find the min and max values for $arg{z}$ in radians to 2dp.

Argument is $tan^{-1}{\frac{12}{5}}$, but not sure where to go to get the max and min values from here?

Thanks in advance

2. b) is correct but you probably need to justify your answer better. After all you can't usually just add and subtract lengths like this.

c) Since the circle does not contain the origin, and does not pass through the x axis the minimum and maximum values of the argument occur at the points where lines through the origin are tangent to the circle. Let the centre of the circle be C and the point of tangency be P. angle OPC is a right angle. You know the length of OC and CP, so you have a right angled triangle for which CP/OC is the sine of angle POC. Given that you already know the argument of C you can then easily find the arguments of the two points of tangency. So you can find the answer without actually finding the coordinates of the two tangent points!

3. Thank you! Didn't think of doing that