Hint
The radius r is positive. Read the fixed complex number in z−c.
Worked solution
(a)
The modulus is the distance to the fixed centre c, so the locus is a circle of radius r.
Answer: \(\left(x - 12\right)^{2} + \left(y - 5\right)^{2} = 16\); centre \(\left( 12, \ 5\right)\), radius \(4\).
(b)
A strict distance inequality gives points less than r from the centre and excludes equality.
Answer: The open interior of the stated circle.
(c)
Square the positive modulus radius.
\(16\)
Answer: \(16\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Translate the modulus distance to a Cartesian equation and identify its centre and positive radius.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The centre is the subtracted complex number c.
(b) · 2 marks
- 1 mark: Use strict distance from the centre to select the open interior and exclude its boundary.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A strict inequality excludes the boundary circle.
(c) · 2 marks
- 1 mark: Square the positive modulus radius.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The centre is the subtracted complex number c.
Total: 6 marks.