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The idea behind your region shading is correct! Just be careful with how you position the lines for the arg locus & the circle for the modulus locus - hint: the arg locus stems from the point (1,1), does this point (1,1) lie inside the circle, or does it lie on the circle?for this graph it was to shade -pi/2 <= arg <= - pi then the bottom greater part would be shaded rightView attachment 45867View attachment 45868
The reason why the Cambridge solution & the Fort St solution is a bit different is because:For the cambridge q, it says that the range of arg(Z) is the same cos its just z being shifted but howcome the same logic doesnt apply to the other q i attached? I understand how the fort st q arrives to the answer but im js confused because the 2 solutions seem a bit contradictoryView attachment 45871View attachment 45872View attachment 45873View attachment 45874
thank you this helps alot!The reason why the Cambridge solution & the Fort St solution is a bit different is because:
- In the Cambridge question, part (b) shifts both the argument and the circle by 2 units to the left, i.e. |z-2|=1 becomes |z|=1, and arg(z) becomes arg(z+2). As a result, the diagram would look exactly like (a), but everything is just moved 2 units to the left.
- In the Fort st question, the circle remains stationary, and only the argument shifts, i.e. the question contains arg(z), arg(z-1), and arg(z-2). As such, the circle remains still, whereas the arguments are constantly moving relative to the circle.
Hope this helps!
wait but why is it the outside part i thought it would be the larger bottom bitThe idea behind your region shading is correct! Just be careful with how you position the lines for the arg locus & the circle for the modulus locus - hint: the arg locus stems from the point (1,1), does this point (1,1) lie inside the circle, or does it lie on the circle?
Sorry I'm not sure if I'm following - what do you mean by this?wait but why is it the outside part i thought it would be the larger bottom bit
Like as in why do u shade the part i attached in the photo and not the region below the 2 perpendicular raysSorry I'm not sure if I'm following - what do you mean by this?
It is because the question is looking for -pi/2 <= arg(z - 1 - i) <= pi, the bit that isn't shaded in your diagram represents -pi <= arg(z - 1 - i) <= -pi/2Like as in why do u shade the part i attached in the photo and not the region below the 2 perpendicular rays