Question

$z$ is a variable complex number such that

$|z|=1$ and $u=3z-\frac{1}{z}$

Show that the locus of the point in the Argand plane representing

$u$ is an ellipse and find the equation of the ellipse.

• A. $(x/2{)}^2+(y/4{)}^2=1$
• B. $x^2/4+y^2/2=1$
• C. $x^2+y^2/4=1$
• D. $x^2/4+y^2=1$

Answer 1

The correct option is A $(x/2{)}^2+(y/4{)}^2=1$

We have $z\overline{z}=|z{|}^2=x^2+y^2=1$. This can also be described as

$z=e^{it}=cos\ t+isin\ t=x(t)+iy(t)$

Now,

$u=3e^{it}-e^{-it}=2cos\left(t\right)+i4sin\left(t\right)=x+iy$

So,

$2cos\left(t\right)=x$

$4sin\left(t\right)=y$

$co{s}^{2}t+si{n}^{2}t=(x/2{)}^2+(y/4{)}^2=1$ is required locus.