If ${\tan ^{ - 1}}\left( {{\rm{\alpha }} + i{\rm{\beta }}} \right) = x + iy$, then $x$ is equal to

$\frac{1}{2}{\tan ^{ - 1}}\left( {\frac{{2{\rm{\alpha }}}}{{1 - {{\rm{\alpha }}^2} - {{\rm{\beta }}^2}}}} \right)$

$\frac{1}{2}{\tan ^{ - 1}}\left( {\frac{{2{\rm{\alpha }}}}{{1 + {{\rm{\alpha }}^2} + {{\rm{\beta }}^2}}}} \right)$

${\tan ^{ - 1}}\left( {\frac{{2{\rm{\alpha }}}}{{1 - {{\rm{\alpha }}^2} - {{\rm{\beta }}^2}}}} \right)$

JEE MATHEMATICS MCQ: COMPLEX NUMBERS

1 Medium

1 of 1

Application +4 | -1

{\tan ^{ - 1}}\left( {\mathrm{\alpha } + i\mathrm{\beta }} \right) = x + iy

, then

x

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}{\tan ^{ - 1}}\left( {\frac{{2\mathrm{\alpha }}}{{1 - {\mathrm{\alpha }^2} - {\mathrm{\beta }^2}}}} \right)

Option A

\frac{1}{2}{\tan ^{ - 1}}\left( {\frac{{2\mathrm{\alpha }}}{{\left( {1 - {\mathrm{\alpha }^2} - {\mathrm{\beta }^2}} \right)}}} \right)

is the correct expression for x, derived by using the tangent addition formula and properties of complex conjugates.