# Finding the locus of a complex number?

##
#z# is a variable complex number such that #|z|=1# and #u=3z-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.

##### 2 Answers

#### Explanation:

We have

Now

or

or

so the ellipse equation is obtained as

Loci is that of an ellipse with equation

# (x/2)^2 + (y/4)^2 =1 #

#### Explanation:

Let

From

# a^2 + b^2 = 1 \ \ \ \ .....[1]#

From

# x+yi = 3(a+bi) - 1/(a+bi) #

# " " = 3a+3bi - 1/(a+bi) * (a-bi)/(a-bi)#

# " " = 3a+3bi - (a-bi)/(a^2-(bi)^2)#

# " " = 3a+3bi - (a-bi)/(a^2+b^2)#

# " " = 3a+3bi - (a-bi) \ \ \ \# (from [1])

# " " = 3a+3bi - a+bi #

# " " = 2a+4bi #

And so:

# x=2a => a=x/2 #

# y=4b => b=y/4 #

Squaring and adding we get:

# a^2 + b^2 = (x/2)^2 + (y/4)^2 #

# :. (x/2)^2 + (y/4)^2 =1 # (from [1])

Which is the equation f an ellipse with semi-minor axis