Let z C, the set of complex numbers.

Question

Let $z\in\mathbb C$, the set of complex numbers. Then the equation $|2z+3i|-|z-i|=0$ represents:

Accepted Answer

$2|z+\tfrac32i|=|z-i|$. With $z=x+iy$:
$$4\left(x^2+\left(y+\frac32\right)^2\right)=x^2+(y-1)^2\ \Longrightarrow\ 3x^2+3y^2+14y+8=0.$$

$$x^2+y^2+\frac{14}3y+\frac83=0,\qquad r^2=\left(\frac73\right)^2-\frac83=\frac{25}9\Rightarrow r=\frac53.$$

A circle of diameter $\dfrac{10}3$.

Solutions · 1