Complex numbers loci

in an argand diagram the loci arg(z-2i)= pi/6 and |z-3|=|z-3i|, intersect at point P. Express complex number represented by P in the form re^(itheta).
I just sketched half line from (0,2) at an angle of pi/6 radians and drew the line (3,0)and (0,3) with perpendicular bisector at(3/2,3/3/2).
I dont know how to do further. Pls explain

Just construct their Cartesian equations and see where they intersect.

Just construct their Cartesian equations and see where they intersect.

The cartesian eq of the perpendicular bisector is y=x. I dont know how to find the cartesian eq of arg(z-2i)= pi/6

Perp. bisector is correct.

For arg(z-2i) = pi/6, this is just a line which (effectively) passes through (0,2) at an angle pi/6 to the horizontal ... you can use this angle to calculate the gradient of the line.

So gradient will be tan pi/6= 1/sqroot 3

Yep.

Yep.

Perp. bisector is correct.

For arg(z-2i) = pi/6, this is just a line which (effectively) passes through (0,2) at an angle pi/6 to the horizontal ... you can use this angle to calculate the gradient of the line.

Yep.

I forgot to type x


So I get x=5 and y=5

The ans in textbook r= 6.69 and iam getting 7 correct to nearest integer is it ok?

x=y=5 is incorrect since you have a sqrt(3) flying about ... I suspect it will stay!

Nope.

Thanks a lott for helping me. Can you also pls help me with one more sum. Its 15(ii)

In effect, it's simply asking you for the shortest distance between the circle and the half-line.

You should try and interpet the question in this way. All the z points lie on the circle, and all the w points lie on the half-line. Then |z-w| is just a distance between a point on the circle, and a point on the half-line.

In effect, it's simply asking you for the shortest distance between the circle and the half-line.

You should try and interpet the question in this way. All the z points lie on the circle, and all the w points lie on the half-line. Then |z-w| is just a distance between a point on the circle, and a point on the half-line.

So do I just take the exact coordinates from the diagram?

The distance will be minimum when the "distance line" is perpendicular to the line. it also passed through the circle centre so that should enable you to find the equation of the distance line and hence the minimum distance.

The calcs should be fairly easy using 45-45-90 side ratios (simple trig).

So the equation of the circle will be x^2+(y+1)^2=1. iam not understanding how to do further