Question

Can someone help with #9 (I’m checking what I got to make sure I did these correct)

Answer 1

Given,

$f(x)=\frac{x+1}{2-x}$

to find,

$f(5i)=\frac{5i+1}{2-5i}$

let us use the conjugate property,

$\begin{gathered} \frac{a+bi}{c+di}\: =\: \frac{\left(c-di\right)\left(a+bi\right)}{\left(c-di\right)\left(c+di\right)} \\ =\frac{\left(ac+bd\right)+\left(bc-ad\right)i}{c^2+d^2} \end{gathered}$

here,

$a=1,b=5,c=2,d=-5$$\begin{gathered} =\frac{\left(1\cdot\:2+5\left(-5\right)\right)+\left(5\cdot\:2-1\cdot\left(-5\right)\right)i}{2^2+\left(-5\right)^2} \\ =\frac{-23+15i}{29} \\ =-\frac{23}{29}+i\frac{15}{29} \end{gathered}$

The answer is,

$-\frac{23}{29}+i\frac{15}{29}$