Question

If the point $z_1=1+i$ where $i=\sqrt{-1}$ is the reflection of point $z_2=x+iy$ in the line $i\overline{z}-iz=5$, then the point $z_2$ is

• A. $1+4i$
• B. $4+i$
• C. $1-i$
• D. $-1-i$

Answer 1

Answer: (A)

$1+4i$

Given, $i\overline{z}-iz=5$

Let, $z=x+iy$ and $\overline{z}=x-iy$

$\therefore i(x-iy)-i(x+iy)=5$

or, $2y-5=\mathrm{0...........}\left(i\right)$

Since, it is given that reflection of point $x+iy,$ i.e. $(x,x)$

about the line (i) is $(1+i)$ i.e. $(1,1)$

$\therefore \frac{1-x}{0}=\frac{1-y}{2}=\frac{-2(2y-5)}{4}$

or, $\frac{1-x}{0}=\frac{-(2y-5)}{2}$ and $\frac{1-y}{2}=\frac{-(2y-5)}{2}$

or, $x=1$ and $1-y=-2y+5$

or, $x=1$ and $y=4$

$\therefore$ Required point is $(1+4i).$

Hence, A is the correct option.