QJEEM22Mad1d7: JEE Main 2022 Mathematics - Class 11 JEEM22Mad1d7 Solution: Let for some real numbers and , . If the system of equations…

Given $a = \alpha - i\beta$ and

$4ix + (1 + i)y = 0$ ...... (i)

$8\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)x + \overline a y = 0$ .... (ii)

By (i)

${x \over y} = {{ - (1 + i)} \over {4i}}$ ...... (iii)

By (ii)

${x \over y} = {{ - \overline a } \over {8\left( {{{ - 1} \over 2} + {{\sqrt 3 i} \over 2}} \right)}}$ ..... (iv)

Now by (iii) and (iv)

${{1 + i} \over {4i}} = {{\overline a } \over {4\left( { - 1 + \sqrt 3 i} \right)}}$

$\Rightarrow \overline a = \left( {\sqrt 3 - 1} \right) + \left( {\sqrt 3 + 1} \right)i$

$\Rightarrow \alpha + i\beta = \left( {\sqrt 3 - 1} \right) + \left( {\sqrt 3 + 1} \right)i$

$\therefore$ ${\alpha \over \beta } = {{\sqrt 3 - 1} \over {\sqrt 3 + 1}} = 2 - \sqrt 3$