Question

$\text{If}\text{the}\text{conjugate}\text{of}\left(\text{x + iy}\right)\left(\text{1 - 2i}\right)\text{be}\text{1 + i}\text{then}\text{x}\text{and}\text{y}\text{are}$

• A. $\frac{3}{5},\frac{4}{5}$
• B. $\frac{3}{5},\frac{1}{5}$
• C. $\frac{-3}{5},\frac{1}{5}$
• D. none of these

Answer 1

The correct option is B $\frac{3}{5},\frac{1}{5}$

Now,

$(x+iy)(1-2i)$

$=(x+iy)-2(-y+ix)$

$=(x+2y)+i(y-2x)$.

So conjugate of this number will be $(x+2y)-i(y-2x)=(x+2y)+i(2x-y)=1+i$ (given).

Then comparing the real and imaginary parts we get,

$x+2y=1$.....(1) and $2x-y=1$.....(2).

Now, (2)$-2\times$ (1) we get,

$-5y=-1$

or, $y=\frac{1}{5}$.

Using this values in (1) we get, $x=1-\frac{2}{5}=\frac{3}{5}$.

So $x=\frac{3}{5},y=\frac{1}{5}$.