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

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

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Standard XII

Mathematics

Purely Real

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 find the value of $Im (z_{1} + z_{2})$ . (where $Im (z)$ denotes the imaginary part of $z$ )

sin \frac{π}{3}

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cosec \frac{π}{3}

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tan \frac{π}{3}

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cot \frac{π}{3}

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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 $z_{2} (\frac{t_{2}^{2} + 1}{2}, t_{2})$
$arg (z_{1} - z_{2}) = \frac{π}{3}$
$\Rightarrow \frac{t_{2} - t_{1}}{(\frac{t_{2}^{2} - t_{1}^{2}}{2})} = \sqrt{3}$
$\Rightarrow \frac{2}{t_{1} + t_{2}} = \sqrt{3}$
$∴ Im (z_{1} + z_{2}) = t_{1} + t_{2} = \frac{2}{\sqrt{3}}$

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

The correct option is B cosec π3Let 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 find the value of $Im (z_{1} + z_{2})$ . (where $Im (z)$ denotes the imaginary part of $z$ )