which values of x and y would make the following expression represent a real number? (4 + 5i)(x + yi) o x =…

which values of x and y would make the following expression represent a real number? (4 + 5i)(x + yi) o x = 4, y = 5 o x = -4, y = 0 o x = 4, y = -5 o x = 0, y = 5

which values of x and y would make the following expression represent a real number? (4 + 5i)(x + yi) o x = 4, y = 5 o x = -4, y = 0 o x = 4, y = -5 o x = 0, y = 5

Answer

Explanation:

Step1: Expand the complex - number product

[ \begin{align*} (4 + 5i)(x+yi)&=4x+4yi+5ix + 5i\times yi\ &=4x+4yi + 5ix-5y\ &=(4x - 5y)+(4y + 5x)i \end{align*} ]

Step2: Set the imaginary part equal to 0

For the complex number ((4x - 5y)+(4y + 5x)i) to be a real number, the imaginary part (4y + 5x) must be 0, i.e., (4y=-5x). Let's check each option:

  • Option 1: If (x = 4,y = 5), then (4y+5x=4\times5 + 5\times4=20 + 20=40\neq0).
  • Option 2: If (x=-4,y = 0), then (4y+5x=4\times0+5\times(-4)=-20\neq0).
  • Option 3: If (x = 4,y=-5), then (4y+5x=4\times(-5)+5\times4=-20 + 20=0).
  • Option 4: If (x = 0,y = 5), then (4y+5x=4\times5+5\times0=20\neq0).

Answer:

C. (x = 4,y=-5)