If complex numbers z 1 and z 2 both satisfy z - z -2- z -1- and z 1- z 2-3- then the value of Im z 1- z 2 isImz denotes the imaginary part of z A- sin-3B- cosec-3C- tan 3D- -3

Let z = x + iy z+z2=|z 1| ⇒ x^2=(x 1)^2+y^2 ⇒ y^2=2x 1 Let z_1(t_1^2+12,t_1) and nbsp;z_2(t_2^2+12,t_2) (z_1 z_2)=π3 ⇒t_2 t_1(t_2^2 t_1^2/2)=√(3) ⇒2t_1+t_2=√( ...

If complex nu...

Question

If complex numbers $z_{1}$ and $z_{2}$ both satisfy $z + ¯ ¯ ¯ z = 2 | z - 1 |$ and $arg (z_{1} - z_{2}) = \frac{π}{3},$ then the value of $Im (z_{1} + z_{2})$ is
( $Im (z)$ denotes the imaginary part of $z$ )

sin \frac{π}{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

cosec \frac{π}{3}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

tan \frac{π}{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

cot \frac{π}{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $cosec \frac{π}{3}$
Let $z = x + i y$
$\frac{z + ¯ ¯ ¯ z}{2} = | z - 1 |$
$\Rightarrow x^{2} = (x - 1)^{2} + y^{2}$
$\Rightarrow y^{2} = 2 x - 1$
Let $z_{1} (\frac{t_{1}^{2} + 1}{2}, t_{1})$ and $z_{2} (\frac{t_{2}^{2} + 1}{2}, t_{2})$
$arg (z_{1} - z_{2}) = \frac{π}{3}$
$\Rightarrow \frac{t_{2} - t_{1}}{(\frac{t_{2}^{2} - t_{1}^{2}}{2})} = \sqrt{3}$
$\Rightarrow \frac{2}{t_{1} + t_{2}} = \sqrt{3}$
$∴ Im (z_{1} + z_{2}) = t_{1} + t_{2} = \frac{2}{\sqrt{3}}$

Suggest Corrections

Similar questions

Q. If complex numbers

z_{1}

and

z_{2}

both satisfy

z + ¯ z = 2 | z - 1 |

and

arg (z_{1} - z_{2}) = \frac{π}{3},

then find the value of

Im (z_{1} + z_{2})

. (where

Im (z)

denotes the imaginary part of

z

)

Q. If complex numbers

z_{1}

and

z_{2}

both satisfy

z + ¯ ¯ ¯ z = 2 | z - 1 |

and

arg (z_{1} - z_{2}) = \frac{π}{3},

then the value of

Im (z_{1} + z_{2})

is
(

Im (z)

denotes the imaginary part of

z

)

Q. If complex numbers

z_{1}

and

z_{2}

both satisfy

z + ¯ z = 2 | z - 1 |

and

arg (z_{1} - z_{2}) = \frac{π}{3},

then find the value of

Im (z_{1} + z_{2})

. (where

Im (z)

denotes the imaginary part of

z

)

Q. If

z_{1}

and

z_{2}

both satisfy

z + ¯ z = 2 | z - 1 |

and

a r g (z_{1} - z_{2}) = \frac{π}{4}

, find

I m (z_{1} + z_{2})

Q. If

z_{1}

and

z_{2}

satisfy

z + ¯ z = 2 | z - 1 |

and

arg (z_{1} - z_{2}) = \frac{π}{4},

then

I m (z_{1} + z_{2}) =

If complex numbers z1 and z2 both satisfy z+¯¯¯z=2|z−1| and arg(z1−z2)=π3, then the value of Im(z1+z2) is (Im(z) denotes the imaginary part of z)

The correct option is B cosec π3 Let z=x+iy z+¯¯¯z2=|z−1| ⇒x2=(x−1)2+y2 ⇒y2=2x−1 Let z1(t21+12,t1) and z2(t22+12,t2) arg(z1−z2)=π3 ⇒t2−t1(t22−t212)=√3 ⇒2t1+t2=√3 ∴Im(z1+z2)=t1+t2=2√3

If complex numbers $z_{1}$ and $z_{2}$ both satisfy $z + ¯ ¯ ¯ z = 2 | z - 1 |$ and $arg (z_{1} - z_{2}) = \frac{π}{3},$ then the value of $Im (z_{1} + z_{2})$ is
( $Im (z)$ denotes the imaginary part of $z$ )