Complex Roots and Loci in an Argand Diagram
A complex number z satisfies |z − 3 − 4i| = 5. A student is asked to describe the locus, find the greatest and least values of |z|, and determine whether z = 8i lies on the locus. The student sketches a circle centred at (3, 4) with radius 5, notes that the circle passes through the origin, and concludes that the least value of |z| is therefore 0. A second part of the task asks for the set of points where arg(z − 3 − 4i) = π/4, and the student claims this is the same circle. The marking guide rewards a clear geometric interpretation and an algebraic verification.
1. Verify the locus and the greatest and least values of |z|, and check whether z = 8i lies on the locus.
Read the worked answer
|z − (3 + 4i)| = 5 is a circle centred at C(3, 4) with radius 5. The distance from the origin to C is √(9 + 16) = 5, so the circle passes through O, confirming the least value of |z| is 0 (at z = 0).
The greatest value of |z| is OC + radius = 5 + 5 = 10, attained at the point on the circle farthest from O, on the ray from O through C: z = 2(3 + 4i) = 6 + 8i.
Check z = 8i: |8i − 3 − 4i| = |−3 + 4i| = √(9 + 16) = 5. So z = 8i does lie on the locus. The student's conclusion about the least value is correct, but the reasoning must show OC = 5 equals the radius rather than asserting it.
2. Describe the locus arg(z − 3 − 4i) = π/4 and explain why it is not the same as the circle. Find any intersection points of the ray with the circle.
Read the worked answer
The condition arg(w) = π/4 with w = z − (3 + 4i) describes a ray from C(3, 4) making angle π/4 with the positive real direction. It is a half-line, not a closed curve, so it cannot coincide with the circle. The ray has equation y − 4 = x − 3 for x ≥ 3, i.e. y = x + 1.
Intersect with the circle (x − 3)² + (y − 4)² = 25. Substituting y = x + 1: (x − 3)² + (x − 3)² = 25, so 2(x − 3)² = 25, giving x − 3 = ±5/√2. Only the positive root keeps x ≥ 3, so x = 3 + 5/√2 ≈ 6.54 and y ≈ 7.54. The single intersection is z ≈ 6.54 + 7.54i. The negative root corresponds to the opposite ray (argument 5π/4), which is excluded by the argument condition.