Forum

A point z moves in ...
 
Notifications
Clear all

A point z moves in the complex plane such that arg (z-2/z+2) = π/4

1 Posts
2 Users
0 Likes
1,041 Views
0
Topic starter

A point z moves in the complex plane such that arg \(\Big(\frac{z-2}{z+2}\Big)\) = \(\frac{\pi}{4}\), then the minimum value of |z - 9√2 - 2i|2 is equal to .........

1 Answer
0

Let z = x + iy

arg \(\Big(\frac{x-2+iy}{x+2+iy}\Big)\)= \(\frac{\pi}{4}\)

arg (x -2 + iy) - arg (x + 2 +iy) = \(\frac{\pi}{4}\)

tan-1 \(\Big(\frac{y}{x-2}\Big)\) - tan-1 \(\Big(\frac{y}{x+2}\Big)\) = \(\frac{\pi}{4}\)

\(\frac{\frac{y}{x-2} - \frac{y}{x+2}}{1+\Big(\frac{y}{x-2}\Big).\Big(\frac{y}{x+2}\Big)}\) = tan\(\frac{\pi}{4}\) = 1

\(\frac{xy + 2y - xy + 2y}{x^2 - 4 + y^2}\) = 1

4y = x2 - 4 + y2

x2 + y2 - 4y - 4 = 0

locus is a circle with center (0, 2) & radius = 2√2

min. value = (AP)2 = (OP – OA)2

= (9√2 - 2√2)2

= (7√2)2 = 98

Share:

How Can We Help?