Q62P - Mathematical Methods in Physical Sciences

Question

Describe geometrically the set of points in the complex plane satisfying the following equations. $\mathbf{|}\mathit{z}\mathbf{+}\mathbf{1}\mathbf{|}\mathbf{+}\mathbf{|}\mathit{z}\mathbf{-}\mathbf{1}\mathbf{|}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{ }$

Step-by-Step Solution

Answer

The equation is of an ellipse.

Step 1: Given Information

The equation is $|z+1|+|z-1|=8$ .

Step 2: Definition of the Complex number.

A complex number can be expressed as:

$\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$

Where z is the complex number, x and y are real numbers, and i is known as iota, whose value is  $\sqrt{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{ }}$ .

The modulus of a complex number can be calculated as:

 $\mathbf{\left|}\mathit{z}\mathbf{\right|}\mathbf{=}\mathbf{\surd }\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{)}$

Step 3: Find the value

The equation is $|z+1|+|z-1|=8$ .

 

The complex number is $\left|\right(1+x)+yi|=8-\left|\right(-1+x)+yi|$ .

$$\begin{gathered} \sqrt{{\left(1+x\right)}^2+y^2}=8- \sqrt{{\left(-1+x\right)}^2+y^2} \\ {\left[\sqrt{{\left(1+x\right)}^2+y^2}\right]}^2={\left[8- \sqrt{{\left(-1+x\right)}^2+y^2}\right]}^2 \\ {\left(1+x\right)}^2+y^2=\left[64-16\sqrt{{\left(-1+x\right)}^2+y^2}+ {\left(1+x\right)}^2+y^2\right] \\ {\left(1+x\right)}^2=\left[64-16\sqrt{{\left(-1+x\right)}^2+y^2}+ {\left(1+x\right)}^2\right] \end{gathered}$$

 

Solve further:

$$\begin{gathered} (1+2x+x^2)=\left[64-16\sqrt{{\left(-1+x\right)}^2+y^2}+ \left(1+2x+x^2\right)\right] \\ \left(4x\right)=\left[64-16\sqrt{{\left(-1+x\right)}^2+y^2}\right] \\ 16\sqrt{{\left(-1+x\right)}^2+y^2}=64-4x \\ 4\sqrt{{\left(-1+x\right)}^2+y^2}=16-x \end{gathered}$$ 

 

Solve further:

 $$\begin{gathered} 16\left[x^2-2x+1+y^2\right]=256-32x+x^2 \\ 15x^2+16y^2=240 \end{gathered}$$

 

Hence, the equation is of an ellipse.