Question

If |z –25i| £ 15, then maximum arg (z) – minimum arg (z) | equals-

• A. $\frac{\pi}{2}$+ cos^–1$\left( \frac{3}{5} \right)$
• B. sin^–1$\left( \frac{3}{5} \right)$– cos^–1 $\left( \frac{3}{5} \right)$
• C. 2 cos^–1 $\left( \frac{4}{5} \right)$
• D. 2 cos^–1$\left( \frac{3}{5} \right)$

Answer 1

Answer: (C)

2 cos^–1 $\left( \frac{4}{5} \right)$

Answer 2

Sol. |z –25i| £ 15

OP = OC² – CP² = 20

sin q =$\frac{20}{25}$= $\frac{4}{5}$, cos q = $\frac{15}{25}$= $\frac{3}{5}$

Minimum amplitude of z is q = sin^–1 $\frac{4}{5}$

maximum amplitude of z is p – q = p – sin^–1 $\frac{4}{5}$

So |max amp. (z) – min. amp. z | = p – q – q

= p – 2q = 2$\left( \frac{\pi}{2} - \theta \right)$= 2$\left\lbrack \frac{\pi}{2} - \sin^{- 1}\frac{4}{5} \right)$= 2 cos^–1 $\frac{4}{5}$