AS 91577 · 2024 paper · a factor, a power in polar form, a surd equation, a locus gradient, and a line a quotient never meets
Question 1(d) Merit
Practises Loci of complex numbers
Five separate parts: a factor of a cubic, a power of a number in polar form, an equation with two square roots, the gradient of a straight-line locus, and a proof that a quotient never lies on y = x.
The locus described by |z − i| = |z + 1| is a straight line.
Find the gradient of that line, and justify your answer.
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 modulus.
|x + (y − 1)i| = |(x + 1) + yi|, so √(x2 + (y − 1)2) = √((x + 1)2 + y2) -
2 Square both sides and expand.
x2 + y2 − 2y + 1 = x2 + 2x + 1 + y2 -
3 Cancel what appears on both sides. What line is left?
−2y = 2x, so y = −x. The gradient is −1. -
4 Does that fit the geometry?
The locus is every point the same distance from i, at (0, 1), and from −1, at (−1, 0): the perpendicular bisector of the segment joining them. That segment has gradient 1, so a line at right angles to it has gradient −1. ✓
What the marker wanted
- Achieved z = x + yi substituted and both sides squared, as in step 2.
- Merit The gradient, −1. The schedule accepts the geometric argument as well.
In our words, from NZQA’s 2024 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.