Question

In the argand¢s plane the locus of z ¹ 1 such that arg$\left\{ \frac{3}{2}\left( \frac{2z^{2} - 5z + 3}{3z^{2}–z–2} \right) \right\}$ = $\frac{2\pi}{3}$ is :

• A. The straight line joining the points z = $\frac{3}{2}$, z = – $\frac{2}{3}$
• B. The straight line joining the points z = – $\frac{3}{2}$, z = $\frac{2}{3}$
• C. A segment of a circle passing through z = $\frac{3}{2}$, z = + $\frac{2}{3}$
• D. A segment of a circle passing through z = + $\frac{3}{2}$, z = – $\frac{2}{3}$

Answer 1

Answer: (D)

A segment of a circle passing through z = + $\frac{3}{2}$, z = – $\frac{2}{3}$

Answer 2

Sol. Here : $\frac{2z^{2} - 5z + 3}{3z^{2} - z - 2}$ = $\frac{(z - 1)(2z - 3)}{(z - 1)(3z + 2)}$

= $\frac{2z - 3}{3z + 2}$ = $\frac{2\left( z - \frac{3}{2} \right)}{3\left( z + \frac{2}{3} \right)}$

Hence, the given condition reduces to ;

arg$\left( \frac{z - 3/2}{z + 2/3} \right)$ = $\frac{2\pi}{3}$, Ž z describes the segment of a circle through z = $\frac{3}{2}$ and z = – $\frac{2}{3}$ at which the chord subtends an angle $\frac{2\pi}{3}$