Question

If (5-3i)(x+iy)/(4-5i) = (2+i)^2-(3-4i)^2, then x=___ and y=____

Answer 1

This looks complicated

lets simplify both sides

By multiplying top and bottom by the conjugate:-

left side  =  (5-3i)(x + iy) (4+5i)          (5-3i)(x + iy) (4+5i) 
                  ------------------------   =      ------------------------
                        (4-5i)(4+5i)                           41

right side =  (2 + i + 3-4)(2+i - 3 + 4i)    (by  Difference of 2 squares)

               =  5-3i)(-1 + 4i)

so as left side = right side

(5-3i)(x+yi)(4+5i)   = 41 (5-3i)(-1+5i)

5-3i is common so:-

(x + iy)(4+5i) = 41(-1+5i)

4x +5xi + 4yi - 5y = -41 + 205i

4x - 5y + (5x + 4y)i = -41 + 205i

EQuating coefficients we are left with the system of equations

4x -  5y = -41
5x + 4y  = 205

solving this gives x = 21 and y = 25