Complex numbers whose real and imaginary parts `x` and `y` are integers and satisfy the equation `3x^(2)-|xy|-2y^(2)+7=0`

Options

A. do not exist
B. exist and have equal modulus
C. form two conjugate pairs
D. do not form conjugate pairs

Answer

B, C

Explanation

`(b,c)` `(i) xy gt 0`, `3x^(2)-xy-2y^(2)=-7` or `(3x+2y)(x-y)=-7`
`x` and `y` being integers, we can take
`3x+2y=7` and `x-y=-1`
`x=1`, `y=2`
If `x` and `y` are charged to `-x`, `-y`, equation remains same.
`x=-1`, `y=-2` is also a solution pair. ,brgt `3x+2y=-1` and `x+y=7` do not give integral solutions.
`(ii) xy=0` will not give any integral solution.
`(iii) xy lt 0 3x^(2)+xy-2y^(2)=-7`
`(3x-2y)(x+y)=-7`
`3x-2y=-7` and `x+y=1` leads to `x=-1` `y=2`
`3x-2y=-7` and `x+y=-1` leads to `x=1` `y=-2`
Required complex numbers are `1+2i`, `1-2i`, `-1+2i`, `-1-2i` which form two conjugate pairs.