Let z₁ and z₂ be two complex numbers such that art(z₁ – z₂) = $\frac{π}{4}$ and z₁, z₂ satisfy the equation $| z - 3 | = R e (z) .$ Then the imaginary part of z₁ + z₂ is equal to______

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alok kumar singh | Contributor-Level 10

2 months ago

Let $z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2} & z = x + i y t h e n$

and $(z_{1} - z_{2}) = \frac{π}{4} g i v e s \frac{y_{1} - y_{2}}{x_{1} - x_{2}} = 1 o r y_{1} - y_{2} = x_{1} - x_{2} . . . . . . . . . . . . (i)$

y² – 6x + 9 = 0 .(ii)

as z₁ & z₂ lies on (ii) so ${y_{1}}^{2} - 6 x_{1} + 9 = 0$ .(iii)

& $y_{2} 2 - 6 x_{2} + 9 = 0 . . . . . . . . . (i v)$

(iii) & (iv) $(y_{1} + y_{2}) (y_{1} - y_{2}) - 6 (x_{1} - x_{2}) = 0$

$(x_{1} - x_{2}) (y_{1} + y_{2} - 6) = 0 f r o m (i)$

-> $y_{1} + y_{2} - 6 = 0 i . e ., y_{1} + y_{2} = 6$

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Let z₁ and z₂ be two complex numbers such that art(z₁ – z₂) = $\frac{π}{4}$ and z₁, z₂ satisfy the equation $| z - 3 | = R e (z) .$ Then the imaginary part of z₁ + z₂ is equal to______

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Let z1 and z2 be two complex numbers such that art(z1 – z2) = π 4 and z1, z2 satisfy the equation | z − 3 | = R e ( z ) . Then the imaginary part of z1 + z2 is equal to______

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Let z₁ and z₂ be two complex numbers such that art(z₁ – z₂) = $\frac{π}{4}$ and z₁, z₂ satisfy the equation $| z - 3 | = R e (z) .$ Then the imaginary part of z₁ + z₂ is equal to______