Please help with this question (1 Viewer)

[hardworker42]

I was doing some revision over the holidays and I came across this question. I have seen another one like it but was unsure how to solve it. Any help would be appreciated

Arg(Z+2) = Pi/4
(b) Fin the modulus and argument of Z when lZl is a minimum
(c) Hence find Z in the form x+iy when lZl is a minimum

Please show all your working and just generally explain how you get the answer.
Thank you

 

[Carrotsticks]

I was doing some revision over the holidays and I came across this question. I have seen another one like it but was unsure how to solve it. Any help would be appreciated

Arg(Z+2) = Pi/4
(b) Fin the modulus and argument of Z when lZl is a minimum
(c) Hence find Z in the form x+iy when lZl is a minimum

Please show all your working and just generally explain how you get the answer.
Thank you

Draw the locus as usual (should be the line y=x+2 for y>0)

What |z| minimum means is the shortest distance of Z from the origin, which is the perpendicular distance from the locus to the origin.

The length of this is the minimum |z|, and find the argument of it as usual.

To find Z in the form x+iy, we know |Z| and the argument of Z, so we can express Z in the form Rcis(theta). Expand this as Rcos(theta) + iRsin(theta) to get the x+iy form (Cartesian Form).

 

[awper7]

The shortest distance is the perpendicular distance, horizontal distance is 2 (From origin to z+2) and arg(z+2) = pi/4.

You can also use trig ratios and angle sum of triangle to solve for the |z| and arg(z). Then put it into mod-arg form. Rcis(theta). Convert to x+iy form.

 

[hardworker42]

Thank you that helped a lot. I just didn't know where to start