Question

If$amp\frac{z-2}{2z+3i}=0$ and $z_0=3+4i$ then

• A. $z_0\overline{z}+\overline{z_0}z=12$
• B. $z_{0}z+\overline{z_0}\overline{z}=12$
• C. $z_0\overline{z}+\overline{z_0}z=0$
• D. none of these

Answer 1

The correct option is B $z_{0}z+\overline{z_0}\overline{z}=12$

Let $z=x+iy$

Then

Imaginary part of $\frac{(z-2)}{2z+3i}=0$

Hence

$\frac{z-2}{2z+3i}$

$=\frac{(x-2)+iy}{2x+i(3+2y)}$

Now imaginary part equals zero.

Hence

$2x\left(y\right)-\left[\right(x-2\left)\right(2y+3\left)\right]=0$

$2xy-[2xy+3x-4y-6]=0$
$4y-3x+6=0$

Or

$3x-4y=6$ ..(i)

Now

$z.z_0$

$(x+iy)(3+4i)$
$=3x-4y+i(3y+4x)$...(a)

And

$\overline{z}.\overline{z_0}$
$=(x-iy)(3-4i)$
$=3x-4y-i(3y+4x)$ ...(b)

Adding a and b, we get

$z{z}_0+\overline{z}.\overline{z_0}$
$=2(3x-4y)$
$=2\left(6\right)$
$=12$.