for what value of the real numbers x and y are the numbers 3 + ix2y and x2 + y - Maths - Complex Numbers and Quadratic Equations

Question

for what value of the real numbers x and y are the numbers 3 +
ix2y and x2 + y + 4i conjugate comp;ex
numbers?????

Pronay · Accepted Answer

Complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs.

$3 + i x^{2} y ...... (1) x^{2} + y + 4 i .....(2)$

Real part of complex number (1) and (2) are equal implies

$x^{2} + y = 3 ...... (3)$

Real part of complex number (1) and (2) are equal and opposite implies

$x^{2} y = - 4 ...... (4)$

Substituting value of y from equation (3) to equation (4) we get

$x^{2} (3 - x^{2}) = - 4 3 x^{2} - x^{4} = - 4 x^{4} - 3 x^{2} - 4 = 0 x^{4} - 4 x^{2} + x^{2} - 4 = 0 x^{2} (x^{2} - 4) + (x^{2} - 4) = 0 (x^{2} + 1) (x^{2} - 4) = 0 \Rightarrow x^{2} - 4 = 0 \Rightarrow x^{2} = 4 \Rightarrow x = \pm 2$

Substituting $x^{2} = 4$ in equation (4) we get

$4 y = - 4 \Rightarrow y = - 1$

Therefore

$x = \pm 2 y = - 1$

for what value of the real numbers x and y are the numbers 3 + ix2y and x2 + y + 4i conjugate comp;ex numbers?????

for what value of the real numbers x and y are the numbers 3 + ix²y and x² + y + 4i conjugate comp;ex numbers?????