Interpreting Modulus Conditions

The given conditions define geometric shapes in the complex plane:

• The equation $|z-6|=5$ represents a circle centered at the complex number $6$ (coordinate $(6, 0)$) with a radius of $5$.
• The equation $|z+2-6i|=5$ represents a circle centered at the complex number $-2+6i$ (coordinate $(-2, 6)$) with a radius of $5$.

Let $z = x + iy$. The equations in Cartesian form are:

1. $(x-6)^2 + y^2 = 5^2 = 25$
2. $(x+2)^2 + (y-6)^2 = 5^2 = 25$

Finding the Intersection Point z

Since both expressions equal 25, we set them equal to each other:

$(x-6)^2 + y^2 = (x+2)^2 + (y-6)^2$

Expand the terms:

$x^2 - 12x + 36 + y^2 = x^2 + 4x + 4 + y^2 - 12y + 36$

Simplify by canceling $x^2$, $y^2$, and $36$ from both sides:

$-12x = 4x + 4 - 12y$

Rearrange the terms to solve for $y$ in terms of $x$:

$12y = 16x + 4$

$3y = 4x + 1 \implies y = \frac{4x+1}{3}$

Substitute this expression for $y$ into the first circle equation:

$(x-6)^2 + \left(\frac{4x+1}{3}\right)^2 = 25$

$x^2 - 12x + 36 + \frac{16x^2 + 8x + 1}{9} = 25$

Multiply the entire equation by 9 to eliminate the fraction:

$9(x^2 - 12x + 36) + (16x^2 + 8x + 1) = 9 \times 25$

$9x^2 - 108x + 324 + 16x^2 + 8x + 1 = 225$

Combine like terms:

$25x^2 - 100x + 325 = 225$

$25x^2 - 100x + 100 = 0$

Divide by 25:

$x^2 - 4x + 4 = 0$

$(x-2)^2 = 0 \implies x = 2$

Substitute $x=2$ back into the expression for $y$:

$y = \frac{4(2)+1}{3} = \frac{8+1}{3} = \frac{9}{3} = 3$

Therefore, the unique complex number satisfying both conditions is $z = 2 + 3i$.

Evaluating the Polynomial Expression

We need to calculate the value of $P(z) = z^3 + 3z^2 - 15z + 141$ for $z = 2 + 3i$.

First, calculate the powers of $z$:

• $z^2 = (2+3i)^2 = 2^2 + 2(2)(3i) + (3i)^2 = 4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12i$
• $z^3 = z \cdot z^2 = (2+3i)(-5+12i) = 2(-5) + 2(12i) + 3i(-5) + 3i(12i) = -10 + 24i - 15i + 36i^2 = -10 + 9i - 36 = -46 + 9i$

Now substitute $z$, $z^2$, and $z^3$ into the polynomial:

$P(2+3i) = (-46 + 9i) + 3(-5 + 12i) - 15(2 + 3i) + 141$

Distribute the constants:

$= -46 + 9i - 15 + 36i - 30 - 45i + 141$

Group the real and imaginary terms separately:

$= (-46 - 15 - 30 + 141) + (9 + 36 - 45)i$

Calculate the sums:

$= (50) + (0)i$

$= 50$

The value of the expression $z^3 + 3z^2 - 15z + 141$ is $50$.