A point z moves in the complex plane such that arg (z2/z+2) = π/4
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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 \(\Big(\frac{z2}{z+2}\Big)\) = \(\frac{\pi}{4}\), 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 \(\Big(\frac{x2+iy}{x+2+iy}\Big)\)= \(\frac{\pi}{4}\)
arg (x 2 + iy)  arg (x + 2 +iy) = \(\frac{\pi}{4}\)
tan^{1} \(\Big(\frac{y}{x2}\Big)\)  tan^{1} \(\Big(\frac{y}{x+2}\Big)\) = \(\frac{\pi}{4}\)
\(\frac{\frac{y}{x2}  \frac{y}{x+2}}{1+\Big(\frac{y}{x2}\Big).\Big(\frac{y}{x+2}\Big)}\) = tan\(\frac{\pi}{4}\) = 1
\(\frac{xy + 2y  xy + 2y}{x^2  4 + y^2}\) = 1
4y = x^{2}  4 + y^{2}
x^{2} + y^{2}  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
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