Questions - Answers

1. if   = 4  determine the locus of z.

Sol:  let z = x₁ + iy₁

                                                ⇒  (x₁ - 3)² + (y₁ + 1)² = 16

                          ⇒ x₁² - 6x₁ + 9 + y₁² + 2y₁ + 1 - 16 = 0.

                          ⇒ x₁² + y₁² - 6x₁ + 2y₁ - 6 = 0

                        ∴ Required locus is x² + y² - 6x + 2y - 6 = 0

2. If z = 2 - 3i,   then show that z² - 4z + 13 = 0

Sol:  Consider z = 2 - 3i    =>   z - 2 = - 3i

                 Squaring on both sides we get

                      (z - 2)² = (-3i)²

                        z² - 4z + 4 = 9i²

                        z² - 4z + 4 = - 9     (∴ i² = -1)

                        z² - 4z + 13 = 0

3. Find the multiplicative inverse of 7 + 24i

Sol: The multiplicative inverse of a + ib is

⇒ conjugate of z₁ is z₂

5. Find the square root of (3 + 4i)

sol: Square root of a +  ib

     
                 Comparing real parts we get

                  x = 1/2  ⇒  2x = 1 ⇒  4x² = 1

                  ⇒ 4x² - 1 =  0

7. Express the complex number into modulus - amplitude form, z = - 1 - i 

Sol:     Given that z = - 1 - i  
           Let z  =  x + iy
            Comparing we get x = - 1,    y = - 
              
           We know that x = r cosθ,   y = r sinθ

 
^._..   cosθ and sinθ are negative, the required angle lies in the third quadrant, so angle is negative.
                                    
 The amplitude of a complex number is known as argument denoted by

                       Arg (z)  =  Arg (x + iy)  =  tan^-1 (y/x)

                       Arg   =  Arg (x - iy)

                Arg (z₁. z₂) = Arg z₁ + Arg z₂ + nπ,  n ∈ {-1, 0, 1}   

                      Arg (Z₁/Z₂) = Arg z₁ - Arg z₂ + nπ, n ∈ {-1, 0, 1}

The sign of argument changes depending on the quadrants accordingly. θ is required argument.

8.  If the Arg and Arg are respectively,  find (Arg z₁ + Arg z₂)

Sol:              Let z₁ = x₁ - iy₁,              =>   

                        So the point lies in the IV quadrant