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# Intersection of two loci (complex numbers) Watch

1. In an argand diagram the two loci:
Arg (z - 2i) = pie/6 and |z - 3| = |z - 3i|
Intersect at a point P. Express this point in the form a + bi.

now I managed to find the equation of the of the second one and it turned out to be be y=x. And i also know that the first loci is a half line passing through 2i at an angle of pie/6 to the horizontal real axis. But how I do I find the point of intersection? Im assuming we somehow need the the line equation both the loci and solve them simultaneously?
2. For the half-line, group the real and imaginary parts, since you know that the tangent of the given angle (pi/6) is equal to the imaginary part over the real part, set up the equation Im(z) / Re(z) = tan(pi/6) and simplify. This will give you a Cartesian equation of the first locus.

Solve the equations simultaneously.

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Updated: July 24, 2014
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