Complex Numbers III: Loci & Multiplication
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
Describe the locus of \(z\) in the Argand diagram for each condition:
(a)\(|z-2|=3\)
(b)\(|z+1-2i|=|z-3|\)
(c)\(rg(z-1)=\dfrac{\pi}{4}\)
(a)
Answer
Circle centre \((2,0)\), radius \(3\)
(b)
Perpendicular bisector
Equal distance from \((-1,2)\) and \((3,0)\)
Answer
Perpendicular bisector of the segment joining \((-1,2)\) and \((3,0)\)
(c)
Answer
Half-line from \((1,0)\) (excluded) at angle \( frac{\pi}{4}\) to the positive real axis
Question 2
Given \(z=2+2i\), find \(z^2\) and \(z^3\) in Cartesian form. Verify using the polar form.
Cartesian
\(z^2=(2+2i)^2=4+8i-4=8i\); \(z^3=z^2\cdot z=8i(2+2i)=16i-16=-16+16i\)
Polar check
\(z=2\sqrt2\, ext{cis} frac{\pi}{4}\); \(z^2=8\, ext{cis} frac{\pi}{2}=8i\) ✓; \(z^3=16\sqrt2\, ext{cis} frac{3\pi}{4}=-16+16i\) ✓
Answer
\(z^2=8i;\;z^3=-16+16i\)
Question 3
Find the locus of \(z\) satisfying \(|z-i|=2|z+1|\). Identify the curve.
Let \(z=x+iy\)
\(\sqrt{x^2+(y-1)^2}=2\sqrt{(x+1)^2+y^2}\)
Square and expand
\(x^2+y^2-2y+1=4(x^2+2x+1+y^2)\Rightarrow-3x^2-3y^2-8x-2y-3=0\Rightarrow x^2+y^2+ frac{8}{3}x+ frac{2}{3}y+1=0\)
Complete the square
\(\left(x+ frac{4}{3}
ight)^2+\left(y+ frac{1}{3}
ight)^2= frac{16}{9}+ frac{1}{9}-1= frac{8}{9}\)
Answer
Circle: centre \(\left(- frac{4}{3},- frac{1}{3}
ight)\), radius \(\dfrac{2\sqrt{2}}{3}\)
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