L3 Complex Numbers
Question Bank
The locus of a complex number $z$ is described by $|z + 4 - 3i| = 5$.
Find the Cartesian equation of the locus, and state the centre and radius of the circle.
The complex number $z$ satisfies $|z - 1 + 2i| \le 3$.
(a) Describe the region of the Argand diagram where $z$ lies, and write it as an inequality in $x$ and $y$, where $z = x + iy$.
(b) Determine whether each of the complex numbers $u = 3$ and $v = 4 - i$ lies in this region.
The locus of a complex number $z$ is described by $|z + 2 - 2i| = |z - 2 - 6i|$.
Find the Cartesian equation of the locus in the form $y = mx + c$, and find the complex number on the locus whose argument is $\dfrac{\pi}{4}$.
The locus described by $|z - 2i| = |z - 4|$ is a straight line.
Find the gradient of this line. Justify your answer.
(a) Describe the locus of $z$ given by $\arg(z - 1) = \dfrac{\pi}{4}$, and give its Cartesian equation, including any restriction on $x$.
(b) Find the complex number(s) $z$ for which $\arg(z - 1) = \dfrac{\pi}{4}$ and $|z| = 5$.
The locus of a complex number $z$ is given by $|z - 3| = |z - 1 - 4i|$.
Find the Cartesian equation of this locus.
Hence, find the value of the real constant $m$ if the complex number $w = m + mi$ lies on this locus.
The locus of a complex number $z$ is described by $|z - 2 - 2i| = 2$.
Find the Cartesian equation of the locus, and the complex number on the locus with the greatest modulus. Give your answer in exact form.
The locus of a complex number $z$ is described by $|z - 7 - 2i| = 2|z - 1 - 2i|$.
Find the Cartesian equation of the locus in the form $(x - a)^2 + (y - b)^2 = r^2$, and state the centre and radius of the circle.
The locus of a complex number $z$ is described by $|2z + 4 - 6i| = 10$.
Find the Cartesian equation of the locus. Hence find the equation of the tangent to the locus at the point where it crosses the positive real axis.
The complex number $w$ is given by $w = \dfrac{z - 8}{z - 6i}$.
Find the Cartesian equation of the locus of $z$ if $\arg(w) = \dfrac{\pi}{2}$, and describe the locus fully.
The locus of a complex number $z$ is described by $|z - 15 + 4i| = 2|z - 6 + 4i|$.
Find the Cartesian equation of the locus. Hence find the least value of $|z|$ for $z$ on the locus.
The locus of a complex number $z$ is described by $|z - 8| = 2|z - 2|$.
Find the Cartesian equation of the locus, and hence find the complex number(s) on the locus with argument $\dfrac{3\pi}{4}$. Give your answer(s) in rectangular form.
The complex number $z$ is not zero, and $w = z + \dfrac{4}{z}$.
Find and describe fully the locus of $z$ if $w$ is a real number.
The locus of a complex number $z$ is described by $|z + 1 - 2i| = |z - 3|$.
Find the Cartesian equation of the locus, and the complex number on the locus that is closest to the origin.
The locus of a complex number $z$ is described by $|z - 3i| + |z + 3i| = 10$.
Find the Cartesian equation of the locus in the form $ax^2 + by^2 = c$, and name the curve. Hence find the complex numbers $z$ on the locus with $\text{Im}(z) = 3$.
Consider the complex numbers $u = 1 + 2ki$ and $v = 2 + 5ki$, where $k$ is a real constant.
Show that the complex number $w = \frac{u}{v}$ will not lie on the line $y = x$ in the Argand diagram, for any value of $k$.
Find the equation of the locus of the point representing $z$ if $|z - 5| - |z + 5| = 6$.
Hence, show that the complex number $w = k(3 + 4i)$, where $k$ is a real constant, does not lie on this locus for any value of $k$.
The locus of a complex number $z$ is described by $|z + 6| = 3|z + 2|$.
The complex number $u = k + i$ lies on this locus.
Find the Cartesian equation of the locus of $z$ (in the form $(x-a)^2 + y^2 = r^2$) and find the possible value(s) of $k$.
The locus of a complex number $z$ is given by the Cartesian equation $y = x + 4$.
Show that the complex number $w = \frac{k + 2i}{1 + ki}$, where $k$ is a real constant, does not lie on this locus for any value of $k$.