If |z|=r and (z)=(π)/(4), then |z+(1)/(z)| equals (a) r+(1)/(r) (b) √(r^2+(1)/(r^2)) (c) √(r+1 /

step 1 In this question it is given that if modulus z is equal to r and argument z is equals to pi by 4

If |z|=r and (z)=(π)/(4), then |z+(1)/(z)| equals (a)...

If $|z|=r$ and $\arg (z)=\frac{\pi}{4}$, then $\left|z+\frac{1}{z}\right|$ equals
(a) $\mathrm{r}+\frac{1}{\mathrm{r}}$
(b) $\sqrt{r^{2}+\frac{1}{r^{2}}}$
(c) $\sqrt{\mathrm{r}+1 / \mathrm{r}}$
(d) $\sqrt{r^{2}+1 / r^{2}-2}$

Key Concepts

Complex Numbers in Polar Form

Complex numbers can be represented in the form re^(i?), where r is the modulus (or absolute value) and ? is the argument (or angle). This representation simplifies many operations such as multiplication, division, and finding powers or roots of complex numbers.

Modulus and Argument

The modulus of a complex number is its distance from the origin in the complex plane, and the argument is the angle it makes with the positive real axis. These two parameters uniquely determine the complex number and are fundamental in converting between polar and rectangular forms.

Reciprocal of a Complex Number in Polar Form

For a complex number expressed as re^(i?), its reciprocal is given by (1/r)e^(-i?). This property follows from the rules of exponents and reverses the original argument, which is useful when performing algebraic operations involving the original complex number and its inverse.

Addition of Complex Numbers

When adding complex numbers, it is often most convenient to convert them to their rectangular form. However, if the complex numbers are expressed in polar form with different magnitudes and/or angles, one may need to use trigonometric identities to combine the real and imaginary parts properly.

Magnitude of a Complex Number

The magnitude (or absolute value) of a complex number expressed as a + bi is calculated using the formula sqrt(a^2 + b^2). This concept is crucial when determining the distance of the complex number from the origin after performing operations such as addition, hence allowing one to find the result in terms of its modulus.

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