Question

If ‘z’ be any complex number such that $\mid 3z-2\mid +\mid 3z+2\mid =4$, then locus of ‘z’ is

• A. A circle
• B. An ellipse
• C. A line segment
• D. None of these

Answer 1

The correct option is C A line segment

Given $\mid 3z-2\mid +\mid 3z+2\mid =4\Rightarrow \mid z-\frac{2}{3}\mid +\mid z+\frac{2}{3}\mid =\frac{4}{3}..........\left(i\right)$

Let $P\left(z\right),A=(\frac{2}{3},0),B=(-\frac{2}{3},0)$then (i) represents $PA+PB=\frac{4}{3}.$

Clearly $AB=\frac{4}{3}\Rightarrow PA+PB=AB.$Thus P is any point on the lines segment AB.