Loci in the Argand diagramAQA A-Level Further Maths: Subtopic test

1. 1
   The complex number $z$ satisfies $|z-3+4\mathrm{i}|=5$.

   (a)

   The locus of $z$ is a circle. Find its centre.

   [1 mark]

   • A$3+4\mathrm{i}$
   • B$3-4\mathrm{i}$
   • C$-3+4\mathrm{i}$
   • D$4-3\mathrm{i}$

   (b)

   Find the greatest value of $|z|$.

   [1 mark]

   • A$5$
   • B$7$
   • C$10$
   • D$25$

   (c)

   Find the Cartesian equation of the locus of $z$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $z$ satisfies $\arg(z+2-\mathrm{i})=\frac{\pi}{4}$.

   (a)

   The locus of $z$ is a half-line. Find the complex number at which it starts.

   [1 mark]

   • A$-2+\mathrm{i}$
   • B$2-\mathrm{i}$
   • C$-2-\mathrm{i}$
   • D$2+\mathrm{i}$

   (b)

   Which of these complex numbers lies on the locus?

   [1 mark]

   • A$1+2\mathrm{i}$
   • B$-4-\mathrm{i}$
   • C$2+3\mathrm{i}$
   • D$3\mathrm{i}$

   (c)

   Find the Cartesian equation of the locus of $z$, stating any restriction on $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $z$ satisfies $|z-2|=|z-2\mathrm{i}|$.

   (a)

   Find the Cartesian equation of the locus of $z$.

   [3 marks]

   (b)

   Find the two complex numbers $z$ that satisfy both $|z-2|=|z-2\mathrm{i}|$ and $|z-2|=2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The region $R$ in the Argand diagram is the set of points $z$ satisfying $|z-3\mathrm{i}|\le 3$ and $\frac{\pi}{4}\le\arg z\le\frac{\pi}{2}$.

   (a)

   (i) Write down the Cartesian equation of the circle $|z-3\mathrm{i}|=3$.
   (ii) The half-line $\arg z=\frac{\pi}{4}$ meets this circle at a point $P$. Find the complex number represented by $P$, explaining why $z=0$ is not a solution.

   [6 marks]

   (b)

   Find the exact area of $R$.

   [6 marks]

   Total for question 4: 12 marks

End of questions