Question

If\[\sqrt{a + ib} = x + iy,\] then possible value of \[\sqrt{a - ib}\] is

Options

• \[x^2 + y^2\]

• \[\sqrt{x^2 + y^2}\]

• x + iy

• x − iy

• \[\sqrt{x^2 - y^2}\]

Answer 1

x\[-\]iy

\[\sqrt{a + ib} = x + iy\]

\[\text { Squaring on both the sides, we get,} \]

\[a + ib = x^2 + (iy )^2 + 2ixy\]

\[ \Rightarrow a + ib = ( x^2 - y^2 ) + 2ixy\]

\[ \therefore a = ( x^2 - y^2 ) \]

\[\text { and } b = 2xy\]

\[ \therefore a - ib = ( x^2 - y^2 ) - 2ixy\]

\[ \Rightarrow a - ib = x^2 + i^2 y^2 - 2ixy \left[ \because i^2 = - 1 \right] \]

\[\text { Taking square root on both the sides, we get: } \]

\[\sqrt{a - ib} = x - iy\]