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19/01/2022 5:00 pm
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Let z1 and z2 be two complex numbers such that arg (z1 – z2) = π/4 and z1, z2 satisfy the equation |z – 3| = Re(z). Then the imaginary part of z1 + z2 is equal to _________.
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19/01/2022 5:12 pm
|z – 3| = Re(z)
let Z = x = iy
⇒ (x – 3)2 + y2 = x2
⇒ x2 + 9 – 6x + y2 = x2
⇒ y2 = 6x – 9
⇒ y2 = 6\((x - \frac{3}{2})\)
⇒ z1 and z2 lie on the parabola mentioned in eq.(1) arg(z1 – z2) = \(\frac{\pi}{4}\)
⇒ Slope of PQ = 1.
Let P\(\Big(\frac{3}{2} + \frac{3}{2} t_1^2, 3t_1\Big)\) and Q\(\Big(\frac{3}{2} + \frac{3}{2} t_2^2, 3t_2\Big)\)
Slope of PQ = \(\frac{3(t_2 - t_1)}{\frac{3}{2}(t_2^1 - t_1^2)}\)
⇒ \(\frac{2}{t_2 + t_2}\) = 1
⇒ t2 + t1 = 2
Im(z1 + z2) = 3t1 +3t2 = 3(t1 +t2) = 3(2)
= 6.00