Given that the complex numbers which satisfy the equation `| z z ^3|+| z z^3|=350` form a rectangle in the Argand plane with the length of its diagonal having an integral number of units, then area of rectangle is 48 sq. units if `z_1, z_2, z_3, z_4` are vertices of rectangle, then `z_1+z_2+z_3+z_4=0` rectangle is symmetrical about the real axis `a r g(z_1-z_3)=pi/4or(3pi)/4`

Options

A. area of rectangle is 48 sq units.
B. if `z_(1),z_(2),z_(3),z_(4)` are vertices of rectangle, then `z_(1)+z_(2)+z_(3)+z_(4)=0`
C. rectangle is symmetrical about the real axis .
D. None of these

Answer

A, B, C

Explanation

Let z = x +iy where x,y satisfy the given equation. Hence , `(x^(2) + y^(2))(x^(2)-y^(2)) = 175`
`rArr x^(2) + y^(2) = 25 and x^(2) -y^(2) = 7` (as all other possibilities will given non-integral solutions)
Hence, possible value of z will be `4+3i,4-3i,-4+3i` and `-4,-3i`.Clearly, it will from a rectangle having lenght of the diagonal 10.

Form the digaram, options (1),(2),(3) are correct.