the point on the curve mod(Z-5i)=3 having the least positive argument, is

To find the point on the curve defined by the equation mod(Z - 5i) = 3 that has the least positive argument, we first need to understand what this equation represents in the complex plane. The expression mod(Z - 5i) = 3 describes a circle centered at the point (0, 5) on the imaginary axis, with a radius of 3.

Visualizing the Circle

In the complex plane, we can represent a complex number Z as Z = x + yi, where x is the real part and y is the imaginary part. The center of our circle, (0, 5), corresponds to the complex number 5i. The radius of 3 means that any point on this circle is 3 units away from the center.

Finding the Points on the Circle

The equation of the circle can be expressed as:

• (x - 0)² + (y - 5)² = 3²

This simplifies to:

• x² + (y - 5)² = 9

To find the points on this circle, we can express y in terms of x:

• (y - 5)² = 9 - x²
• y - 5 = ±√(9 - x²)
• y = 5 ± √(9 - x²)

Determining the Argument

The argument of a complex number Z = x + yi is given by the angle θ formed with the positive real axis, calculated as:

• θ = arctan(y/x)

To minimize the positive argument, we need to find the point on the circle that is closest to the positive real axis. This occurs when the x-coordinate is maximized, as the argument approaches zero when y is small relative to x.

Finding the Relevant Points

Since the center of the circle is at (0, 5) and the radius is 3, the circle intersects the real axis at two points:

• When y = 5 + 3 = 8, we have the point (0, 8).
• When y = 5 - 3 = 2, we have the point (0, 2).

However, we also need to consider the points where the circle intersects the line y = 0, which gives us the points (±3, 0) since these are the horizontal extremes of the circle.

Calculating the Arguments

Now, let’s calculate the arguments for the points of interest:

• For the point (3, 0): θ = arctan(0/3) = 0.
• For the point (-3, 0): θ = arctan(0/-3) = π (not positive).
• For the point (0, 8): θ = arctan(8/0) = π/2.
• For the point (0, 2): θ = arctan(2/0) = π/2.

Identifying the Minimum Positive Argument

The point (3, 0) has an argument of 0, which is the least positive argument. However, since we are looking for the least positive argument that is not zero, we can consider the points (0, 8) and (0, 2), both of which yield an argument of π/2. Thus, the point on the curve with the least positive argument is:

The point (3, 0) on the circle mod(Z - 5i) = 3 has the least positive argument of 0.