Question Number 226562 by thetpainghtun_111 last updated on 04/Dec/25
$$\mathrm{If}\:\mathrm{x}\:+\:\mathrm{y}\:{i}\:=\:\frac{\mathrm{a}\:+\:{i}}{\mathrm{a}\:−\:{i}}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{ay}\:−\:\mathrm{1}\:=\:\mathrm{x}. \\ $$$$\:\left(\mathrm{x}+\mathrm{y}{i}\right)\left(\mathrm{a}−{i}\right)=\mathrm{a}+{i} \\ $$$$\:\:\:\mathrm{ax}\:−\:\mathrm{x}{i}\:+\:\mathrm{ay}{i}\:−\:\mathrm{y}{i}^{\mathrm{2}} \:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\left(\mathrm{ax}\:+\:\mathrm{y}\right)\:+\:\left(\mathrm{ay}\:−\:\mathrm{x}\right){i}\:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\mathrm{ay}\:−\:\mathrm{x}\:=\:\mathrm{1} \\ $$$$\:\:\mathrm{x}\:=\:\mathrm{ay}\:−\mathrm{1} \\ $$
Answered by MrAjder last updated on 06/Dec/25
![∀t∈N,a=t,b=t,c=2t,d=3t^2 a^4 +b^4 +c^4 =t^4 +t^4 +(2t)^4 =18t^4 =2(3t^2 )^2 =2d^2 [Q.E.D]](https://www.tinkutara.com/question/Q226574.png)
$$\forall{t}\in\mathbb{N},{a}={t},{b}={t},{c}=\mathrm{2}{t},{d}=\mathrm{3}{t}^{\mathrm{2}} \\ $$$${a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{c}^{\mathrm{4}} ={t}^{\mathrm{4}} +{t}^{\mathrm{4}} +\left(\mathrm{2}{t}\right)^{\mathrm{4}} =\mathrm{18}{t}^{\mathrm{4}} =\mathrm{2}\left(\mathrm{3}{t}^{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{2}{d}^{\mathrm{2}} \\ $$$$\left[{Q}.{E}.{D}\right] \\ $$