The point on the circle |z-5 i|=4 having the greatest...

The point on the circle $|z-5 i|=4$ having the greatest argument is
(a) $\left(\frac{-1}{5}, \frac{9}{5}\right)$
(b) $\left(\frac{-2}{5}, \frac{9}{5}\right)$
(c) $\left(\frac{-12}{5}, \frac{9}{5}\right)$
(d) $\left(\frac{-12}{5}, \frac{3}{5}\right)$

Key Concepts

Complex Numbers

Complex numbers are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. They can be represented graphically on the complex plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part, allowing for a geometric interpretation of operations such as addition, multiplication, and finding magnitudes and angles.

Argument of a Complex Number

The argument of a complex number is the angle measured from the positive real axis to the line representing the complex number in the complex plane. This angle, typically measured in radians, indicates the direction of the vector corresponding to the complex number and is crucial when transforming between rectangular and polar forms.

Circle in the Complex Plane

A circle in the complex plane is defined by an equation of the form |z - z?| = r, where z is a complex number, z? is a fixed complex number representing the center of the circle, and r is the radius. This geometric representation facilitates the analysis of complex relationships and transformations, linking algebraic expressions with geometric shapes.

Geometric Optimization

Geometric optimization involves finding a point on a given geometric object (such as a circle) that maximizes or minimizes a particular quantity, like the argument of a complex number. This requires an understanding of both the geometric properties of the shape and the analytical criteria defining the quantity to be optimized.

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