Question

If \[ |z-3i|+|z+5i|=4, \] then the locus of \(z\) is:

• A. No such point \(z\) exists
• B. Ellipse
• C. Parabola
• D. Circle

Hint:

For expressions of the form \(|z-z_1|+|z-z_2|\), compare the given sum with the distance between the fixed points \(z_1\) and \(z_2\). If the sum is smaller, no locus exists.

Answer 1

The Correct Option is A

Step 1: Interpret the modulus expressions as distances.
Let z = x + iy. The term |z - 3i| measures the distance from z to the fixed point (0, 3). Similarly, |z + 5i| = |z - (-5i)| measures the distance from z to the point (0, -5).

Step 2: Calculate the separation between the two fixed points.
The points are (0, 3) and (0, -5). The straight-line distance between them is |3 - (-5)| = 8.

Step 3: Apply the triangle inequality.
For any complex number z, the sum of its distances to two fixed points cannot be smaller than the distance between those points themselves. Hence |z - 3i| + |z + 5i| ≥ 8. However, the given condition demands this sum equal 4. Since 4<8, the requirement is geometrically impossible.

Step 4: Final conclusion.
No complex number z can satisfy the given equation. Therefore, no such point z exists.