Question

The equation $\left| {z - {z_ \circ }} \right| = r$ represents
(Given: ${z_ \circ }$ is a fixed complex number)
A.A line
B.A circle with centre z0​ and radius r
C.A circle with centre (0,0) and radius 1
D.A line through origin

Answer 1

Here we need to determine whether the given equation is of line or circle. We will first assume the fixed complex number which is present in the equation and the variable complex number of the equation. Then we will substitute these values in the equation and then we will solve the equation and from there, we will get the required answer.

Complete step-by-step answer:
The given equation is $\left| {z - {z_ \circ }} \right| = r$. We need to find the nature of $z$.
Here it is given that ${z_ \circ }$ is a fixed complex number, which means that this complex number represents only a single point on the complex plane.
Here $z$ is a variable complex number.
Let ${z_ \circ } = {x_ \circ } + i{y_ \circ }$ , where ${x_ \circ }$ and ${y_ \circ }$ are constants and
$z = x + iy$, where $x$ and $y$ are constants.
Now, we will substitute these values in the given equation.
$\Rightarrow \left| {x + iy - {x_ \circ } - i{y_ \circ }} \right| = r$
Subtracting real part from real part and imaginary part from imaginary part, we get
$\Rightarrow \left| {\left( {x - {x_ \circ }} \right) + i\left( {y - {y_ \circ }} \right)} \right| = r$
On simplifying the modulus of this complex number, we get
$\Rightarrow \sqrt {{{\left( {x - {x_ \circ }} \right)}^2} + {{\left( {y - {y_ \circ }} \right)}^2}} = r$
On squaring both sides, we get
$\Rightarrow {\left( {x - {x_ \circ }} \right)^2} + {\left( {y - {y_ \circ }} \right)^2} = {r^2}$
We know that this equation represents the equation of a circle with center as $\left( {{x_ \circ },{y_ \circ }} \right)$ and the radius equal to $r$
Since, the point $\left( {{x_ \circ },{y_ \circ }} \right)$ is represented by the complex number ${z_ \circ }$, so we can say that the center of the circle is ${z_ \circ }$.
Hence, the correct option is option B.

Note: Since we have used complex numbers here and also we have solved the complex equations, so we need to know its definitions. A complex number is defined as the number which is the combination of both real number and an imaginary number and the complex number is represented by $a + ib$ .