Let z1 and z2 be two complex numbers such that z1 - z2 - -4 and z1- z2 satisfy the equation -z-3- - Rez- Then the imaginary part of z1 - z2 is equal to

Z_1 = x_1+ iy_1 and z_2 = x_2 + iy_2 and z_1 z_2 = (x_1 x_2) + 2(y_1 y_2) (z_1 z_2) = π4 ⇒tan^ 1( y_1 y_2x_1 x_2) = π4 ⇒ y_1 y_2 = x_1 x_2⋯(i) ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Argument of a Complex Number

Let z1 and z2...

Question

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $arg (z_{1} - z_{2}) = \frac{π}{4}$ and $z_{1}, z_{2}$ satisfy the equation $| z - 3 | = R e (z)$ . Then the imaginary part of $z_{1} + z_{2}$ is equal to

Open in App

Solution

$z_{1} = x_{1} + i y_{1}$ and $z_{2} = x_{2} + i y_{2}$ and $z_{1} - z_{2} = (x_{1} - x_{2}) + 2 (y_{1} - y_{2})$

$arg (z_{1} - z_{2}) = \frac{π}{4} \Rightarrow {tan}^{- 1} (\frac{y_{1} - y_{2}}{x_{1} - x_{2}}) = \frac{π}{4}$
$\Rightarrow y_{1} - y_{2} = x_{1} - x_{2} \dots (i)$
$\Rightarrow | z - 3 | = R e (z) \Rightarrow | (x - 3) + 2 y | = x$
$\Rightarrow (x - 3)^{2} + (y)^{2} = x^{2}$
$\Rightarrow y^{2} = 6 (x - \frac{3}{2})$
Let the point on this parabola
$(\frac{3}{2} + a t_{1}^{2}, 2 a t_{1})$ and $(\frac{3}{2} + a t_{2}^{2}, 2 a t_{2}),$ where $a = \frac{6}{4}$
$∵ y_{1} - y_{2} = x_{1} - x_{2}$
$\Rightarrow 2 a (t_{1} - t_{2}) = a (t_{1}^{2} - t_{2}^{2})$
$\Rightarrow t_{1} + t_{2} = 2$
Now, $I m (z_{1} + z_{2}) = y_{1} + y_{2}$
$= 2 a (t_{1} + t_{2})$
$= 2 \times \frac{6}{4} (2) = 6$

Suggest Corrections

Similar questions

Q. Let

z_{1}

z_{2}

be complex numbers such that

z_{1} \neq z_{2}

| z_{1} | = | z_{2} |

. If

z_{1}

has positive real part and

z_{2}

has negative imaginary part, then

\frac{z_{1} + z_{2}}{z_{1} - z_{2}}

may be

Q. Let

z_{1}

and

z_{2}

be complex numbers such that

z_{1} \neq z_{2}

and

| z_{1} | = | z_{2} |

. If

z_{1}

has positive real part and

z_{2}

has negative imaginary part, then

(z_{1} + z_{2}) / (z_{1} - z_{2})

may be

Q. Let

Z_{1}, Z_{2}

be two complex numbers such that

Z_{1} \neq Z_{2}

and

| Z_{1} | = | Z_{2} |

. lf

Z_{1}

has positive real part,

Z_{2}

has negative imaginary part then

\frac{Z_{1} + Z_{2}}{Z_{1} - Z_{2}} i s

Q. Let

z_{1}

and

z_{2}

be complex numbers such that

z_{1} \neq z_{2}

and

| z_{1} | = | z_{2} |

. If

z_{1}

has positive real part and

z_{2}

has negative imaginary part, then

\frac{z_{1} + z_{2}}{z_{1} - z_{2}}

may be

Q. If

z_{1}

and

z_{2}

be complex numbers such that

z_{1} \neq z_{2}

and

| z_{1} | = | z_{2} |

. If

z_{1}

has a positive real part and

z_{2}

has negative imaginary part, then

\frac{z_{1} + z_{2}}{z_{1} - z_{2}}

may be

Join BYJU'S Learning Program

Let z1 and z2 be two complex numbers such that arg(z1−z2)=π4 and z1,z2 satisfy the equation |z−3|=Re(z). Then the imaginary part of z1+z2 is equal to

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $arg (z_{1} - z_{2}) = \frac{π}{4}$ and $z_{1}, z_{2}$ satisfy the equation $| z - 3 | = R e (z)$ . Then the imaginary part of $z_{1} + z_{2}$ is equal to