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Attempt following question by selecting a choice to answer. D Determine the solution for the equation

|Z| - Z = 8 + 9i . DDDD DDDD A.

D 17 16 - 9

i B.

D 9 +

17 16 i C.

D 9 -

17 16 i D.

D 17 16 + 9

i DD
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: x ² + y ² = x + iy + 8 + 9i [Substitute |Z| = x ² + y ² , Z = x + iy .] Step 4: x ² + y ² = (x + 8) + i (y + 9) Step 5: x ² + y ² = (x + 8)² + 2i (x + 8) (y + 9) + i ²(y + 9)²[Square on both the sides.] Step 6: x ² + y ² = [x ² + 16x + 64 - y ² - 18y - 81] + i [2(x + 8) (y + 9)] Step 7: y ² = [16x + 64 - y ² - 18y - 81] + i [2(x + 8) (y + 9)]
[Simplify.] Step 8: y ² = 16x + 64 - y ² - 18y - 81[Equate the real part.] Step 9: (x + 8) (y + 9) = 0 ⇒ x = - 8, y = - 9[Equate the imaginary part.] Step 10: y ² = - 128 + 64 - y ² - 18y - 81[Substitute x = - 8 in Step 8.] Step 11: 2y ² + 18y + 145 = 0 Step 12: y = - 1 8 ± 3 2 4 - 1 1 6 0 4 (imaginary)[Apply formula for roots of the quadratic equation ax ² + bx + c = 0.] Step 13: 81 = 16x + 64 - 81 + 162 - 81[Substitute y = - 9 in Step 8.] Step 14: 16x = 17 ⇒ x = 17 16 Step 15: Z = 17 16 - 9i [Substitute x = 17 16 , y = - 9 in Z = x + iy .] Step 16: Therefore, the solution for the equation is 17 16 - 9i