If a complex number has arg(z) = \(\pi\over3\) and the product of real and imaginary part is 3√3, find the conjugate of the complex number z.

Concept:

Let z = x + iy be a complex number.

• Modulus of z = \(\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im\;}}{{\left( {\rm{z}} \right)}^2}}\)
• arg (z) = arg (x + iy) = \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)
• Conjugate of z = x - iy

Calculation:

Let z = x + iy

According to the question 

arg(z) = \(\rm \tan^{-1}{y\over x} = {\pi\over3}\)

⇒\(\rm {y\over x} = \tan{\pi\over3}\)

⇒ \(\rm {y\over x} = \sqrt3\) 

⇒ y = \(\sqrt3\)x

Also given xy = 3\(\sqrt3\)

⇒ y = \(\rm3\sqrt3\over x\)

Comparing the values of y, we get

⇒ \(\rm{3\sqrt3\over x} = \sqrt3x\)

⇒ x² = 3

⇒ x = ± \(\sqrt3\)

∴ x = \(\sqrt3\) {∵ arg(z) lies in first quadrant so -ve value is neglected}

Also y = \(\sqrt3\)x = 3

Complex number z = x + iy = \(\sqrt3\) + 3i

Conjugate of z = (z̅) = \(\boldsymbol {\sqrt3}\) - 3i