Loci and regions in the Argand diagramEdexcel International A Level Further Maths: Subtopic test

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

   (a)

   Which statement describes the locus of $z$?

   [1 mark]

   • AA circle with centre $-3+4i$ and radius $5$
   • BA circle with centre $3-4i$ and radius $5$
   • CA circle with centre $3-4i$ and radius $25$
   • DA circle with centre $3+4i$ and radius $5$

   (b)

   Which of these complex numbers lies on the locus?

   [1 mark]

   • A$3$
   • B$3-4i$
   • C$-3$
   • D$6$

   (c)

   Find the greatest value of $|z|$ for points on the locus.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $z=x+iy$ satisfies $|z-2|=|z+4i|$.

   (a)

   Which statement describes the locus of $z$?

   [1 mark]

   • AA circle with centre $1-2i$ and radius $\sqrt5$
   • BThe perpendicular bisector of the points representing $2$ and $4i$
   • CThe perpendicular bisector of the points representing $2$ and $-4i$
   • DThe straight line through the points representing $2$ and $-4i$

   (b)

   Which is the Cartesian equation of the locus?

   [1 mark]

   • A$x+2y+3=0$
   • B$x+2y-3=0$
   • C$2x+y+3=0$
   • D$x-2y+3=0$

   (c)

   Find, in terms of $x$ and $y$, the inequality satisfied by points in the region $|z-2|\le|z+4i|$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $z$ satisfies $\arg(z-2+i)=\frac\pi3$, and $L$ is the locus of $z$.

   (a)

   Describe $L$ geometrically, and find its Cartesian equation.

   [3 marks]

   (b)

   Find the complex number on $L$ whose real part equals its imaginary part.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $z=x+iy$ be a complex number, and consider the complex numbers $-3$ and $6$.

   (a)

   Show that the locus of points satisfying $|z+3|=\frac12|z-6|$ is a circle, and find its centre and radius.

   [6 marks]

   (b)

   Find the Cartesian equation of the locus $\arg\left(\dfrac{z+3}{z-6}\right)=\dfrac\pi2$, and state which part of the curve it is.

   [6 marks]

   Total for question 4: 12 marks

End of questions