If z satisfies |(z-12)/(z-8 i)|=(5)/(3) and |(z-4)/(z-8)|=1, then |z| is equal to (a) √(3) (b) 1

step 1 In this question we need to find the value of modulus z if z satisfies modulus z minus 12 upon z

If z satisfies |(z-12)/(z-8 i)|=(5)/(3) and |(z-4)/(z-8)|=1,...

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part, typically expressed in the form z = a + bi. They are fundamental in many areas of mathematics, providing a way to work with quantities that have both magnitude and direction in the complex plane.

Modulus of a Complex Number

The modulus of a complex number is a measure of its distance from the origin in the complex plane. It is defined as |z| = ?(a² + b²) for a number z = a + bi, and it plays a crucial role in understanding the size and spread of complex numbers in various problems.

Locus of Points in the Complex Plane

A locus in the complex plane is a set of points that satisfy a specific condition or equation. Equations involving complex numbers often define geometric figures like lines or circles, where each point in the figure meets the given criteria.

Apollonius Circle

An Apollonius circle is a specific type of locus in the complex plane defined by an equation of the form |(z - a)/(z - b)| = k, where a and b are fixed complex numbers and k is a positive constant not equal to 1. This circle represents all points z for which the ratio of distances to the two fixed points is constant.

Intersection of Loci

In problems involving multiple equations, the solution often lies at the intersection of different loci, such as lines and circles. Analyzing these intersections involves solving the equations simultaneously, which typically leads to finding the common points that satisfy all the given conditions.

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