AS 91577 · 2024 paper · a quotient in terms of k, equal roots, a modulus from a quotient, a complex cubic, and a circle locus
Question 2(e) Merit to Excellence
Practises Loci of complex numbers
Five separate parts: a quotient written as a + bi, a quadratic with equal roots, the modulus of an unknown found from a quotient, a cubic with one complex root given, and a locus that is a circle.
The locus of a complex number z is described by |z − 1 − 7i| = 2|z − 4 − 4i|. The complex number u = 3 + di lies on this locus.
Find the Cartesian equation of the locus of z, giving your answer in the form (x − a)2 + (y − b)2 = k, and also find the complex number(s) u.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Put z = x + yi into each side and write out each modulus.
|(x − 1) + (y − 7)i| = 2|(x − 4) + (y − 4)i|
√((x − 1)2 + (y − 7)2) = 2√((x − 4)2 + (y − 4)2) -
2 Square both sides. The 2 becomes 4.
(x − 1)2 + (y − 7)2 = 4[(x − 4)2 + (y − 4)2] -
3 Expand and collect everything on one side.
x2 − 2x + y2 − 14y + 50 = 4x2 − 32x + 4y2 − 32y + 128
0 = 3x2 − 30x + 3y2 − 18y + 78, or x2 − 10x + y2 − 6y + 26 = 0 -
4 Complete the square in x and in y.
(x − 5)2 − 25 + (y − 3)2 − 9 + 26 = 0, so (x − 5)2 + (y − 3)2 = 8: a circle, centre 5 + 3i, radius 2√2. -
5 u = 3 + di is on it. Put x = 3 and y = d in.
(3 − 5)2 + (d − 3)2 = 8, so (d − 3)2 = 4, d − 3 = ±2, and d = 5 or d = 1.
What the marker wanted
- Achieved Both sides squared correctly after z = x + yi (step 2).
- Merit Expanded and collected into one equation with all its terms on one side (step 3).
- Excellence The circle in the form asked for AND both values of u: E8. Both values of d found but not written as the two numbers u, or a full answer with one minor error: E7.
In our words, from NZQA’s 2024 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.