Question

Locate the points representing the complex number z on the argand plane:
$\left| z+1-2i \right|=\sqrt{7}$

Answer 1

Hint: First we will compare it with all the 2nd degree equations and after finding what this equation represents we will try to draw it on the graph to understand it even better. To plot a graph of a circle we will need the centre and radius of the circle.

Complete step-by-step answer:

First let’s write the given equation in some different form and then we will try to understand the true meaning of the equation.

$\left| z+1-2i \right|=\sqrt{7}$

$\left| z-(-1+2i) \right|=\sqrt{7}\text{ }............\text{ (1)}$

Now if we compare it with the general equation of circle which is:

$\left| z-{{z}_{0}} \right|=r\text{ }...........\text{ (2)}$ , here ${{z}_{0}}$ = centre of circle and r = radius.

If we look carefully equation (1) and (2) are similar,

From this we can conclude that the given equation is the equation of the circle.

Now we know that it is a circle and we can find it’s centre and radius.

Centre = ( -1 + 2i ) and radius = $\sqrt{7}$ .

Now we know everything to plot a graph.

Let’s see what it looks like:

Here x axis represents the real axis and y axis represents imaginary axis.

As we see in the above figure that point C is the centre of the circle and the green line is the desired points that satisfy the given equation.

Note: There is another method to solve this question, one can convert the variables in x and y by putting z = x + iy and then if we solve the equation further then it will represent the equation of circle with centre at (-1,2) and now we can draw the graph in the Cartesian coordinate.