Question:

**Step 1: Convert to Standard Form** Let $z = x + iy$. Then $\bar{z} = x - iy$ and $z\bar{z} = x^2 + y^2$. Substitute these into the given equation: $$x^2 + y^2 + (-4 + 5i)(x - iy) + (-4 - 5i)(x + iy) - 40 = 0$$ **Step 2: Expand the Complex Terms** Expand the products: $$(-4 + 5i)(x - iy) = -4x + 4iy + 5ix - 5i^2y = -4x + 5y + i(5x + 4y)$$ $$(-4 - 5i)(x + iy) = -4x - 4iy - 5ix - 5i^2y = -4x + 5y + i(-5x - 4y)$$ Summing these two expressions: $$(-4x + 5y + i(5x + 4y)) + (-4x + 5y + i(-5x - 4y)) = -8x + 10y + i(5x + 4y - 5x - 4y) = -8x + 10y$$ **Step 3: Separate Real and Imaginary Parts** Substitute the sum back into the equation from Step 1: $$x^2 + y^2 - 8x + 10y - 40 = 0$$ **Step 4: Complete the Square** Rearrange the terms and complete the square for $x$ and $y$: $$(x^2 - 8x) + (y^2 + 10y) = 40$$ $$(x^2 - 8x + 16) + (y^2 + 10y + 25) = 40 + 16 + 25$$ $$(x - 4)^2 + (y + 5)^2 = 81$$ This equation represents a circle in the complex plane with center $C = 4 - 5i$ and radius $r = \sqrt{81} = 9$. **Step 5: Find Distance from Center to Reference Point** We need to find the maximum and minimum values of $|z + 2 - 3i|$. This can be rewritten as $|z - (-2 + 3i)|$. Let $P = -2 + 3i$ be the reference point. The distance $d$ between the center $C = 4 - 5i$ and the point $P = -2 + 3i$ is: $$d = |P - C| = |(-2 + 3i) - (4 - 5i)| = |-2 - 4 + 3i - (-5i)| = |-6 + 8i|$$ $$d = \sqrt{(-6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$ **Step 6: Calculate Maximum and Minimum** The maximum value $a$ of $|z + 2 - 3i|$ occurs when $z$ is on the circle and furthest from $P$. This is $d + r$. $$a = d + r = 10 + 9 = 19$$ The minimum value $b$ of $|z + 2 - 3i|$ occurs when $z$ is on the circle and closest to $P$. This is $d - r$. $$b = d - r = 10 - 9 = 1$$ **Step 7: Calculate the required values** Using $a=19$ and $b=1$: $$a + b = 19 + 1 = 20$$ $$a^2 + b^2 = 19^2 + 1^2 = 361 + 1 = 362$$ $$a - b = 19 - 1 = 18$$ $$ab = 19 \times 1 = 19$$