Question

If z1 / z2 is a pure fantasy number then prove it

$|z _{1}+z _{2}| {}^{2} = |z _{1} | {}^{2} + |z _{2}| {}^{2} ,$

Where x and y are non-complex composite numbers

Answer 1

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$\bold{\underline{Question-}}$

If z1 / z2 is a pure fantasy number then prove it

$|z _{1}+z _{2}| {}^{2} = |z _{1} | {}^{2} + |z _{2}| {}^{2} ,$

Where x and y are non-complex composite numbers



$\bold{\underline{Answer-}}$



Let, $z _{1} = x + iy$ And, $z_{2} = a + ib$

$\frac{ z_{1} }{ z_{2}} = \frac{x + iy}{a + ib} \\ \\ \\ = \frac{x + iy}{a + ib} + \frac{a - ib}{a - ib} \\ \\ \\ = \frac{ ax + by + i(ay + bx)}{ {a}^{2} + {b}^{2} }$


if z1/z2 is pure fantasy number, then ax+by =0

$|z _{1}+z _{2}| {}^{2} = |x + iy + a + ib| {}^{2} \\ \\ = |(x + a) + i(y + b)| {}^{2} \\ \\ = {(x + a)}^{2} + {(y + b)}^{2} \\ \\ = {x}^{2} + {y}^{2} + {a}^{2} + {b}^{2} + 2(ax + by) \\ \\ = ( {x}^{2} + {y}^{2} ) + ( {a}^{2} + {b}^{2} )$

[ °•° ax + by = 0 ]


$|z _{1} | {}^{2} + |z _{2}| {}^{2} \:$