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

1. 1
   A point $P$ representing the complex number $z$ moves so that $|z-3+2i|=4$.

   (a)

   What are the coordinates of the centre of the circle traced by $P$?

   [1 mark]

   • A$(3,2)$
   • B$(-3,-2)$
   • C$(3,-2)$
   • D$(-3,2)$

   (b)

   Which of these complex numbers lies on the locus?

   [1 mark]

   • A$7-2i$
   • B$3-2i$
   • C$7+2i$
   • D$4-6i$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A point $P$ representing the complex number $z$ moves so that $|z-1|=|z-i|$.

   (a)

   Which description of the locus of $P$ is correct?

   [1 mark]

   • AA circle with centre at the midpoint of $1$ and $i$
   • BThe perpendicular bisector of the line segment joining the points representing $1$ and $i$
   • CThe straight line through the points representing $1$ and $i$
   • DA circle with centre $1$ and radius $|i|$

   (b)

   What is the Cartesian equation of the locus?

   [1 mark]

   • A$y=-x$
   • B$y=1-x$
   • C$y=x+1$
   • D$y=x$

   (c)

   The point $P$ also satisfies $|z|=4$. Find the exact possible values of $z$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The locus $L$ of a point $P$ representing the complex number $z$ is given by $\arg(z-2i)=\frac{\pi}{4}$.

   (a)

   Describe the locus $L$ geometrically and find its Cartesian equation.

   [3 marks]

   (b)

   The complex number $w$ lies on $L$ and satisfies $|w-2i|=2\sqrt2$. Find $w$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Sets of points in an Argand diagram are defined by $S_1=\{z:|z-3-4i|\le5\}$ and $S_2=\{z:|z|\le|z-6|\}$.

   (a)

   Describe $S_1$ and $S_2$ geometrically. Find the complex numbers represented by the points where the boundaries of $S_1$ and $S_2$ intersect.

   [6 marks]

   (b)

   Show that the points representing $0$ and $6$ lie on the boundary of $S_1$, and find the exact area of $S_1\cap S_2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions