Question

Let z = x + iy, the number of integral ordered pairs (x, y) for which z2 + |z|2 = 0, if x E (-4, 4) and y E [-4, 4] is (where i = -1​

Answer 1

SOLUTION

TO DETERMINE

Let z = x + iy, the number of integral ordered pairs (x, y) for which z² + |z|² = 0

where x ∈ ( - 4 , 4 ) and y ∈ [- 4 , 4 ]

EVALUATION

Here the given complex number is

z = x + iy

We get from above

$\sf{ |z| = \sqrt{ {x}^{2} + {y}^{2} } }$

Now the given equation is

$\sf{ {z}^{2} + { |z| }^{2} = 0 }$

$\sf{ \implies {(x + iy)}^{2} + { | \sqrt{ {x}^{2} + {y}^{2} } | }^{2} = 0 }$

$\sf{ \implies {x }^{2} + 2ixy + {i}^{2} {y}^{2} + {x}^{2} + {y}^{2} = 0 }$

$\sf{ \implies {x }^{2} + 2ixy - {y}^{2} + {x}^{2} + {y}^{2} = 0 }$

$\sf{ \implies 2{x }^{2} + 2ixy = 0 }$

Comparing real part and Imaginary parts in both sides we get

$\sf{ {x}^{2} = 0 \: \: \: \: and \: \: 2xy = 0}$

$\sf{ \implies x = 0 \: \: \: \: and \: \: \: xy = 0}$

As y ∈ [- 4 , 4 ] from above we get the solution set by taking the integer values of y as below

• x = 0 , y = - 4

• x = 0 , y = - 3

• x = 0 , y = - 2

• x = 0 , y = - 1

• x = 0 , y = 0

• x = 0 , y = 1

• x = 0 , y = 2

• x = 0 , y = 3

• x = 0 , y = 4

Consequently the values of z are

- 4i , - 3i , - 2i , - i , 0 , i , 2i , 3i , 4i

Hence the required number of number of integral ordered pairs is 9

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