Question

given that w=1+2i , z=(w-25(1-2i))/w^2.find value of z, where z is complex number.

Answer 1

Step-by-step explanation:

We have,

$\mathscr{z = \dfrac{ \omega - 25(1 - 2i)}{ {\omega}^{2} } }$

$\mathscr{ \implies \: z = \dfrac{ \omega - 25 + 50i}{ {\omega}^{2} } }$

$\mathscr{ \implies \: z = \dfrac{1 + 2i- 25 + 50i}{ \left(1 + 2i \right)^{2} } }$

$\mathscr{ \implies \: z = \dfrac{ - 24 + 52i}{ 1 + 4 {i}^{2} + 4i } }$

$\mathscr{ \implies \: z = \dfrac{ - 24 + 52i}{ 1 - 4 + 4i } }$

$\mathscr{ \implies \: z = \dfrac{ - 24 + 52i}{ -3 + 4i } }$

$\mathscr{ \implies \: z = \dfrac{ 24 - 52i}{ 3 - 4i } }$

$\mathscr{ \implies \: z = \dfrac{ \left(24 - 52i \right)\left( 3 + 4i \right)}{ \left( 3 - 4i \right)\left( 3 + 4i \right) } }$

$\mathscr{ \implies \: z = \dfrac{ 72 - 156i + 96i - 208 {i}^{2} }{ \left( 3 \right)^{2} + \left( 4 \right)^{2} } }$

$\mathscr{ \implies \: z = \dfrac{ 72 - 156i + 96i + 208 }{ 9 + 16 } }$

$\mathscr{ \implies \: z = \dfrac{280 - 60i }{ 25 } }$

$\mathscr{ \implies \: z = \dfrac{280 }{ 25 } - \dfrac{60}{25}i }$

$\mathscr{ \implies \: z = \dfrac{56 }{ 5 } - \dfrac{12}{5}i }$