Final answer:
The equation |z-4|+|z+4|=16 in the complex plane describes an ellipse with foci at the points -4 and 4 on the real axis, and major axis length 16. By letting z = x + yi and manipulating the absolute value conditions in Cartesian coordinates, we obtain the standard form of the Cartesian equation for an ellipse.
Step-by-step explanation:
The student has provided the condition for a locus in the complex plane: |z−4|+|z+4|=16. This type of problem is located within the field of complex numbers and geometric representations of equations in the complex plane. To find the Cartesian equations of this locus, we have to interpret the condition as the sum of distances from any point z on the complex plane to two fixed points, which in this case are -4 and 4 on the real axis.
From geometry, this is a definition of an ellipse, where the two fixed points are focal points. Here's a step-by-step approach to find the Cartesian equation:
- Let z = x + yi, where x and y are real numbers, and i is the imaginary unit.
- Substitute z into the given condition: |(x + yi) - 4| + |(x + yi) + 4| = 16.
- The absolute value of a complex number a + bi is √(a² + b²), so apply this to both terms.
- You get √((x - 4)² + y²) + √((x + 4)² + y²) = 16.
- To find the Cartesian equation, square both sides of the equation once, and then isolate and square the remaining radical to eliminate the square roots, keeping in mind the properties of a quadratic equation.
- After simplification, you will obtain the standard form of the Cartesian equation for an ellipse.
This process reveals that the locus described by the complex condition is an ellipse centered along the x-axis with foci at (-4, 0) and (4, 0) and major axis length 16.