Question:
If argument of $\frac{z – z_1}{z-z_2}$ is $\pi\over4$, find the locus of $z$.
$$z_1 = 2 + 3i$$$$z_2 = 6 + 9i$$
Approach:
I tried to solve the equation using diagram, basically plotting the points on the Argand plane. What I got is a circle with center $7 + 4i$ and a radius of $\sqrt{26}$ units. The two complex numbers given lie on this circle, and form a chord. Any point lying on the major arc of this chord satisfies the condition.
How exactly would I represent this as a locus of the point?
And is there any other method that I can use that does not involve a diagram?
Best Answer
the angle subtended by the chord $z_1z_2$ at the center is $2 \pi/4 = \pi/2$ so the radius is $\frac{|z_1-z_2|}{\sqrt 2} = \sqrt{26}$ the center of the chord is $4 + 3i$ you add or subtract $\dfrac{-6+4i}{2}$ so that you will get two centers. the two centres, $z_1$ and $z_2$ form a square of side $\sqrt{ 26}.$