I am sorry if i posted in the wrong subforum
here is the question, i know to draw the graph but duno how to do part ii)
any help?
thanks!!
$\displaystyle |z| = |z - 4i|$
$\displaystyle |x + iy| = |x + i(y - 4)|$
$\displaystyle \sqrt{x^2 + y^2} = \sqrt{x^2 + (y - 4)^2}$
$\displaystyle x^2 + y^2 = x^2 + (y - 4)^2$
$\displaystyle y^2 = (y - 4)^2$
$\displaystyle y^2 = y^2 - 8y + 16$
$\displaystyle 0 = -8y + 16$
$\displaystyle 8y = 16$
$\displaystyle y = 2$.
You also know that $\displaystyle \tan{\theta} = \frac{y}{x}$ and $\displaystyle \theta = \frac{\pi}{6}$.
Putting it together, with $\displaystyle y = 2$ gives
$\displaystyle \tan{\frac{\pi}{6}} = \frac{2}{x}$
$\displaystyle \frac{1}{\sqrt{3}} = \frac{2}{x}$
$\displaystyle \frac{x}{\sqrt{3}} = 2$
$\displaystyle x = 2\sqrt{3}$.
So therefore, the point of intersection is
$\displaystyle 2\sqrt{3} + 2i$.
You should know the geometric interpretation of these two loci:
$\displaystyle |z - z_1| = |z - z_2|$ defines the perpendicular bisector of the line segment joining $\displaystyle z_1$ and $\displaystyle z_2$.
$\displaystyle Arg(z - z_1) = \theta$ defines a ray with terminus $\displaystyle z_1$ (not included in the locus because Arg(0) is not defined) and inclined at an angle $\displaystyle \theta$ to the horizontal.
Just an additional point of view. For future reference, try to visualize the geometry beforehand using the geometric meaning of the equations. For example, your first equation states that |z - 0| = |z - 4i|. This means that the distance between z and 0 is the same as the distance between z and 4i. This is describing the horizontal line of points with imaginary part 2 (right between 0 and 4i).
The second equation was covered by the previous poster as a ray. You can then find the intersection point by using basic trigonometry, as the ray, horizontal line, and imaginary axis form a right triangle.
While you can work it out algebraically, it is often much faster to see the full geometric picture first, using the concepts of distance and angle in the complex plane as analogues to modulus and argument, respectively. Of course, this only helps if you have a solid grounding in plane geometry as well (ie., you should know the conics as geometric descriptions of loci, not just their Cartesian equations).