Chapter 2: Q13 MP (page 80)

Find real $x$ and $y$ for which $| z + 3 | = 1 - i z$ , where $z = x + i y$ .

Short Answer

Expert verified

The real values of $(x, y) = (0, 4)$ .

Step by step solution

Given Information

The given equation is $|z + 3| = 1 - i z$ .

Definition of Complex Numbers

The numbers that are presented in the form of $x + i y$ where, $' x'$ is real numbers and $' i y'$ is an imaginary number, those numbers are referred to as called Complex numbers.

Find the ratio of the series.

Consider, $|z + 3| = 1 - i z$ . …(1)

Let $z = x + i y$ and put in equation (1).

$|x + y i + 3| = 1 - (x - y i) |x + y i + 3| = (1 + y) - x i$ ..(2)

Compare both the real and imaginary parts of the equation (2).

The real part is $|x + y i + 3| = (1 + y)$ . ....(3)

The imaginary part is $- x i = 0$ . …(4)

Use equation (4) to get the value of $x = 0$ .

Put the value $x = 0$ in equation (3).

$|y i + 3| = (1 + y) |y i + 3| = \sqrt{y^{2} + 3^{2}}$

Compare both the above equation.

$\sqrt{y^{2} + 3^{2}} = 1 + y y^{2} + 3^{2} = {(1 + y)}^{2} y^{2} + 3^{2} = 1 + 2 y + y^{2} y = 4$

Hence the real values of $(x, y) = (0, 2)$ .

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Find real $x$ and $y$ for which $| z + 3 | = 1 - i z$ , where $z = x + i y$ .

Short Answer

Step by step solution

Given Information

Definition of Complex Numbers

Find the ratio of the series.

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Find real xand yfor which|z+3|=1-iz, where z=x+iy.

Short Answer

Step by step solution

Given Information

Definition of Complex Numbers

Find the ratio of the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

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Magnetism

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Study anywhere. Anytime. Across all devices.

Find real $x$ and $y$ for which $| z + 3 | = 1 - i z$ , where $z = x + i y$ .