An Example on - -

Solved Examples
 Attempt following question by selecting a choice to answer. cDetermine the solution for the equation |Z| - Z = 8 + 9i.cc   ccA.  c1716 - 9iB.  c9 + 1716iC.  c9 - 1716iD.  c1716 + 9ic A Step 1: Let Z = x + iy[Standard form of a complex number.]Step 2: |Z| - Z = 8 + 9i ⇒ |Z| = Z + 8 + 9i[Add Z on both the sides.]Step 3: x²+y² = x + iy + 8 + 9i[Substitute |Z| = x²+y², Z = x + iy.]Step 4: x²+y² = (x + 8) + i (y + 9)Step 5: x² + y² = (x + 8)² + 2i(x + 8) (y + 9) + i²(y + 9)²[Square on both the sides.]Step 6: x² + y² = [x² + 16x + 64 - y² - 18y - 81] + i[2(x + 8) (y + 9)]Step 7: y² = [16x + 64 - y² - 18y - 81] + i[2(x + 8) (y + 9)] [Simplify.]Step 8: y² = 16x + 64 - y² - 18y - 81[Equate the real part.]Step 9: (x + 8) (y + 9) = 0 ⇒ x = - 8, y = - 9[Equate the imaginary part.]Step 10: y² = - 128 + 64 - y² - 18y - 81[Substitute x = - 8 in Step 8.]Step 11: 2y² + 18y + 145 = 0Step 12: y = - 18±324-11604 (imaginary)[Apply formula for roots of the quadratic equation ax² + bx + c = 0.]Step 13: 81 = 16x + 64 - 81 + 162 - 81[Substitute y = - 9 in Step 8.]Step 14: 16x = 17 ⇒ x = 1716Step 15: Z = 1716 - 9i[Substitute x = 1716, y = - 9 in Z = x + iy.]Step 16: Therefore, the solution for the equation is 1716 - 9i