Question

Let \( z \) be a complex number such that \( |z-6|=5 \) and \( |z+2-6i|=5 \). Then the value of \( z^3+3z^2-15z+14 \) is equal to:

• A. \(37\)
• B. \(42\)
• C. \(50\)
• D. \(61\)

Hint:

When two circles with equal radii touch externally, the common point lies exactly at the midpoint of their centers.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The given equations represent two circles in the complex plane.
The intersection of these circles will provide the possible values for \(z\).
After finding \(z\), we substitute it into the polynomial expression, or better yet, use the quadratic equation satisfied by \(z\) to simplify the expression.
Step 2: Key Formula or Approach:
Circle 1: \(|z - 6| = 5\), Center \(C_1 = (6, 0)\), Radius \(r_1 = 5\).
Circle 2: \(|z - (-2 + 6i)| = 5\), Center \(C_2 = (-2, 6)\), Radius \(r_2 = 5\).
Distance between centers \(d = \sqrt{(-2-6)^2 + (6-0)^2} = \sqrt{64 + 36} = 10\).
Since \(r_1 + r_2 = 5 + 5 = 10\), the circles touch externally.
Step 3: Detailed Explanation:
The point of contact \(z\) is the midpoint of the line segment joining the centers.
\[ z = \frac{6 + (-2 + 6i)}{2} = \frac{4 + 6i}{2} = 2 + 3i \] From \(z = 2 + 3i\), we have \(z - 2 = 3i\).
Squaring both sides:
\[ (z - 2)^2 = (3i)^2 \Rightarrow z^2 - 4z + 4 = -9 \Rightarrow z^2 - 4z + 13 = 0 \] Now, we divide the given polynomial \(P(z) = z^3 + 3z^2 - 15z + 141\) by \(z^2 - 4z + 13\).
Using long division or synthetic substitution:
\[ z^3 + 3z^2 - 15z + 141 = z(z^2 - 4z + 13) + 7z^2 - 28z + 141 \] \[ = z(0) + 7(z^2 - 4z + 13) + 50 = 0 + 0 + 50 \] Step 4: Final Answer:
The value of the expression is 50.