Question:

Step 1: Analyze the given complex number equation. The given equation is $\left|\dfrac{z-2i}{z+i}\right|=2$. We use the property of modulus that $\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}$. Applying this, we get: $$ \frac{|z-2i|}{|z+i|} = 2 $$ Multiplying both sides by $|z+i|$, we obtain: $$ |z-2i| = 2|z+i| $$ Step 2: Substitute $z = x+iy$ into the equation. Let $z = x+iy$, where $x$ and $y$ are real numbers. Substitute this into the equation from Step 1: $$ |x+iy-2i| = 2|x+iy+i| $$ Group the real and imaginary parts: $$ |x + i(y-2)| = 2|x + i(y+1)| $$ Step 3: Apply the definition of the modulus of a complex number. The modulus of a complex number $a+ib$ is given by $\sqrt{a^2+b^2}$. Applying this to both sides of the equation: $$ \sqrt{x^2 + (y-2)^2} = 2\sqrt{x^2 + (y+1)^2} $$ Step 4: Square both sides of the equation. To eliminate the square roots, square both sides of the equation: $$ x^2 + (y-2)^2 = (2\sqrt{x^2 + (y+1)^2})^2 $$ $$ x^2 + (y-2)^2 = 4(x^2 + (y+1)^2) $$ Step 5: Expand and simplify the equation. Expand the squared terms and distribute the 4 on the right side: $$ x^2 + (y^2 - 4y + 4) = 4(x^2 + (y^2 + 2y + 1)) $$ $$ x^2 + y^2 - 4y + 4 = 4x^2 + 4y^2 + 8y + 4 $$ Step 6: Rearrange the terms to form the standard equation of a circle. Move all terms to one side of the equation to simplify: $$ 0 = 4x^2 - x^2 + 4y^2 - y^2 + 8y + 4y + 4 - 4 $$ $$ 0 = 3x^2 + 3y^2 + 12y $$ Step 7: Divide by 3 to simplify further. Divide the entire equation by 3: $$ x^2 + y^2 + 4y = 0 $$ Step 8: Complete the square for the $y$ terms. To find the center and radius, we rewrite the equation in the standard form of a circle $(x-h)^2 + (y-k)^2 = r^2$. We complete the square for the $y$ terms by adding and subtracting $(4/2)^2 = 4$: $$ x^2 + (y^2 + 4y + 4) - 4 = 0 $$ $$ x^2 + (y+2)^2 = 4 $$ This can be written as: $$ (x-0)^2 + (y-(-2))^2 = 2^2 $$ Step 9: Identify the center and radius of the circle. Comparing the equation $(x-0)^2 + (y-(-2))^2 = 2^2$ with the standard form $(x-h)^2 + (y-k)^2 = r^2$, we can identify the center $(h,k)$ and the radius $r$: The center of the circle is $(0, -2)$. The radius of the circle is $2$. Step 10: State the final answer. The complex number $z$ lies on a circle of radius 2 and center $(0, -2)$. This matches Option 4. The final answer is $\boxed{\text{$(0,-2)$}}$.