Chapter 2: Q62P (page 55)

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

Short Answer

Expert verified

The equation is of an ellipse.

Step by step solution

Given Information

The equation is $| z + 1 | + | z - 1 | = 8$ .

Definition of the Complex number.

A complex number can be expressed as:

$z = x + i y$

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

The modulus of a complex number can be calculated as:

$| z | = \sqrt (x^{2} + y^{2})$

Find the value

The equation is $| z + 1 | + | z - 1 | = 8$ .

The complex number is $| (1 + x) + y i | = 8 - | (- 1 + x) + y i |$ .

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

Solve further:

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

Solve further:

$16 [x^{2} - 2 x + 1 + y^{2}] = 256 - 32 x + x^{2} 15 x^{2} + 16 y^{2} = 240$

Hence, the equation is of an ellipse.

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

Short Answer

Step by step solution

Given Information

Definition of the Complex number.

Find the value

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Describe geometrically the set of points in the complex plane satisfying the following equations.|z+1|+|z-1|=8.

Short Answer

Step by step solution

Given Information

Definition of the Complex number.

Find the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Astrophysics

Magnetism and Electromagnetic Induction

Quantum Physics

Dynamics

Magnetism

Study anywhere. Anytime. Across all devices.

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