Indicate the regions in the complex plane represented by (i) |z+1|^2+|z-1|^2=4 (ii) (π)/(6) ≤z ≤

Indicate the regions in the complex plane represented by (i)...

Key Concepts

Complex Modulus

This concept refers to the absolute value of a complex number, representing its distance from the origin in the complex plane. It is defined as |z| for a complex number z = x + iy, where |z| = ?(x² + y²), and is crucial for expressing distances from points other than the origin by using expressions such as |z - a|, which represents the distance between z and another fixed complex number a.

Locus Equations in the Complex Plane

Locus equations involve sets of points that satisfy a given equation or inequality, often representing familiar geometric figures such as circles, lines, or angles. When complex moduli appear in equations (for example, the sum of squares of distances from given points equals a constant), algebraic manipulation usually leads to standard forms like that of a circle, thus allowing one to identify the geometric region described.

Complex Argument

The argument of a complex number is the angle that the line connecting the origin to the point makes with the positive real axis. Denoted arg z, it is fundamental in representing complex numbers in their polar form and understanding rotational and angular properties of complex numbers.

Sector Regions in the Complex Plane

Sector regions are obtained by imposing inequalities on the argument of a complex number. This results in the collection of points lying between two rays (or lines) that originate at the origin, defining a wedge or sector. Such regions are important in problems that restrict the direction of the vector corresponding to the complex number.

Logarithmic Inequalities

Logarithmic inequalities in the context of complex analysis (or real analysis as applied to moduli) involve comparing logarithms of expressions where the base of the logarithm plays a crucial role. Understanding whether the base is greater than one or between zero and one determines the direction of the inequality when the logarithm is removed, and transforming these inequalities into algebraic form requires careful handling of these properties.

Transformation of Complex Conditions to Real Inequalities

Often, complex expressions, particularly those involving moduli and logarithms, are converted into equivalent real inequalities by expressing the complex variable in terms of its modulus and, when necessary, its argument. This step is vital to simplify and solve the inequalities by using standard algebraic techniques, thereby revealing the nature of the region in the complex plane defined by the original conditions.

Transcript

00:01 In this question, we need to indicate the regions in the complex frame represented by the given equations.

00:08 In the first part, the given equation is modulus z plus 1 to the power 2 plus modulus z minus 1 to the power 2 is equal to 4.

00:21 Let's see how to solve this question.

00:24 We know that z is the complex number which is represented by x plus iota y.

00:32 Therefore, we can write our equation as modulus x plus iota y plus 1 to the power 2 plus modulus x plus iota y minus 1 to the power 2 is equal to 4.

00:54 This will be equal to x plus 1 to the power 2 plus y square plus x minus 1 to the power 2 plus y square plus x minus 1 to the 2 plus y square is equal to 4.

01:10 On further solving we get 2 x square plus y square is equal to 2 hence this will be equal to x square plus y square is equal to 1.

01:27 This is a equation of circle and this circle is shown below.

01:37 So this is our circle.

01:38 This is x -axis, this is y -axis, and this circle is represented by motorist z is equals to 1.

01:48 And z lies on the unit circle, on the unit circle.

02:05 Now come to part second, pi by 6 less than equals to argument z less than equals to pi by 4.

02:31 This reason is shown in the figure given below.

02:38 This is the figure.

02:40 This line represents pi by 6 and this line is at an angle of pi by 4.

02:53 The reason between these two lines, this reason is the required reason...