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07/02/2022 12:04 pm
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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 .........
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07/02/2022 12:18 pm
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