Any complex number x + iy can be put in the form r( cosq + isinq) where r is called the modulus and q is the argument of the complex number. The complex number z having the least positive argument and satisfying | z – 5i | £ 3 is ?
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complex number (1 Viewer)
- Thread starter rks
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First plot the region given in the argand plane. Draw a line from the origin as a tangent to the circle in the first quadrant. The point of contact represents the requred complex number. Use geometry to determine the co-ordinates from this point. ( a bit tricky, but try marking in the angle theta, and showing it is the same as the angle subtended by the y axis and normal to the tangent (drawn in previously), at the centre of the circle. then its a matter of pythagoras and trig. )
Trev
stix
I think he means '£' which would be either be equal to/greater than/less than to make sense.
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By my understanding of the question, Vafa has it pretty well covered. However I think it only wants the one solution where arg(z) is a minimum positive. By pythagoras as you can see in Vafa's diagram, the modulus here will be 4. I found it more useful to express the argument in a cleaner form, like tan inverse (4/3), seeing as arguments are usually put in this form. so my answer was z = 4(cos(tan inverse (4/3)) + isin(tan inverse (4/3)) ). Using a triangle you can express it as z=4(cos(cos inverse(3/5) + isin(sin inverse(4/5)) ie z= 12/5 + 16i/5