If z=x+i y, determine the Cartesian equation of the locus of the point z which moves in the Arga

step 1 Question number 18, Z is equal to x plus j y. Z plus j2 is equal to x plus j into y plus 2 or z

If z=x+i y, determine the Cartesian equation of the locus of...

Key Concepts

Complex Number Representation

A complex number is written in the form z = x + iy, where x and y are real numbers, representing the real and imaginary parts respectively. This representation allows one to view complex numbers as points or vectors in the two-dimensional coordinate system known as the Argand diagram, where x is the horizontal axis and y is the vertical axis.

Modulus (Absolute Value) of a Complex Number

The modulus of a complex number, given by |z| = ?(x² + y²), measures the distance of the point z from the origin in the complex plane. Squaring the modulus, |z|², directly gives the square of this distance, thereby simplifying many algebraic manipulations in problems involving distances.

Distance Formula in the Cartesian Plane

The distance formula, derived from the Pythagorean theorem, is used to calculate the distance between two points in the Cartesian plane, and is essential for converting complex modulus expressions to Cartesian equations. It is expressed as the square root of the sum of the squares of the differences in the x and y coordinates.

Translation in the Complex Plane

Adding a constant complex number to z corresponds to a translation of the point in the complex plane. For instance, when z is shifted by a pure imaginary number, it moves vertically, which directly affects its y-coordinate while leaving the x-coordinate unchanged. This transformation is useful for analyzing the geometric loci of complex numbers after such shifts.

Cartesian Equation of a Circle

A Cartesian equation of a circle centered at the origin is given by x² + y² = r², where r is the radius of the circle. Recognizing such an equation is crucial in identifying geometric loci from algebraic expressions involving squared moduli of complex numbers, as seen in problems where the sum of squared distances leads to a circular locus.

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