Consider a complex number z on the argand plane satisfying argz2−ω2=π2+argz2−ω where ω=ei 2π3 . If minimum value ofz−2−2i z+2+2i is a−b2a,b∈N then find the value of a+b52 .

Book Online Demo

Check Your IQ

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

Consider a complex number z on the argand plane satisfying $\arg (z^{2} - ω^{2}) = \frac{π}{2} + \arg (z^{2} - ω) (w h e r e ω = e^{\frac{i 2 π}{3}})$ . If minimum value of $|z - 2 - 2 i| |z + 2 + 2 i| i s (\frac{\sqrt{a} - \sqrt{b}}{2}) (a, b \in N)$ then find the value of $(\frac{a + b}{52})$ .

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

answer is 5.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

$\arg (\frac{z^{2} - ω^{2}}{z^{2} - ω}) = \frac{π}{2}$ $z^{2}$
lies on a circle whose center is $(\frac{- 1}{2}, 0)$ and radius is equal to $\frac{\sqrt{3}}{2}$ units.
$|z - 2 (1 + i)| |z + 2 (1 + i)| = |z^{2} - 4 (2 i)| = |z^{2} - 8 i|$
Minimum value of $|z^{2} - 8 i|$ is equal to $\sqrt{\frac{1}{4} + 64} - \frac{\sqrt{3}}{2} = \frac{\sqrt{257} - \sqrt{3}}{2} \equiv \frac{\sqrt{a} - \sqrt{b}}{2}$ $∴ \frac{a + b}{\sqrt{2}} = \frac{257 + 3}{52} = 5$