Question Number 123114 by nimnim last updated on 23/Nov/20 | ||
$${Is}\:{there}\:{any}\:{solution}\left({s}\right)?? \\ $$$$\begin{cases}{\mathrm{36}{x}^{\mathrm{2}} {y}−\mathrm{27}{y}^{\mathrm{3}} =\mathrm{8}}\\{\mathrm{4}{x}^{\mathrm{3}} −\mathrm{27}{xy}^{\mathrm{2}} =\mathrm{4}}\end{cases} \\ $$$${please}.... \\ $$ | ||
Commented by benjo_mathlover last updated on 23/Nov/20 | ||
$$\:\begin{cases}{\mathrm{36}{x}^{\mathrm{2}} {y}−\mathrm{27}{y}^{\mathrm{3}} =\mathrm{8}}\\{\mathrm{8}{x}^{\mathrm{3}} −\mathrm{54}{xy}^{\mathrm{2}} =\mathrm{8}}\end{cases} \\ $$$$\Rightarrow\mathrm{36}{x}^{\mathrm{2}} {y}−\mathrm{8}{x}^{\mathrm{3}} +\mathrm{54}{xy}^{\mathrm{2}} −\mathrm{27}{y}^{\mathrm{3}} =\mathrm{0} \\ $$$$\Rightarrow{x}^{\mathrm{2}} \left(\mathrm{36}{y}−\mathrm{8}{x}\right)+\mathrm{27}{y}^{\mathrm{2}} \left(\mathrm{2}{x}−{y}\right)=\mathrm{0} \\ $$$$\Rightarrow{x}^{\mathrm{2}} \left(\mathrm{36}{y}−\mathrm{8}{x}\right)=\mathrm{27}{y}^{\mathrm{2}} \left({y}−\mathrm{2}{x}\right) \\ $$$$\Rightarrow\left(\frac{{x}}{{y}}\right)^{\mathrm{2}} \left(\mathrm{36}−\frac{\mathrm{8}{x}}{{y}}\right)=\mathrm{27}\left(\mathrm{1}−\mathrm{2}\left(\frac{{x}}{{y}}\right)\right) \\ $$$${let}\:\frac{{x}}{{y}}\:=\:{r}\: \\ $$$$\Rightarrow{r}^{\mathrm{2}} \left(\mathrm{36}−\mathrm{8}{r}\right)=\mathrm{27}\left(\mathrm{1}−\mathrm{2}{r}\right) \\ $$$$\Rightarrow\mathrm{36}{r}^{\mathrm{2}} −\mathrm{8}{r}^{\mathrm{3}} \:=\:\mathrm{27}−\mathrm{54}{r} \\ $$$$\Rightarrow\mathrm{8}{r}^{\mathrm{3}} −\mathrm{36}{r}^{\mathrm{2}} −\mathrm{54}{r}+\mathrm{27}=\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{2}{r}+\mathrm{3}\right)\left(\mathrm{4}{r}^{\mathrm{2}} −\mathrm{24}{r}+\mathrm{9}\right)=\mathrm{0} \\ $$$${r}_{\mathrm{1}} \:=\:−\frac{\mathrm{3}}{\mathrm{2}}\Rightarrow\frac{{x}}{{y}}=−\frac{\mathrm{3}}{\mathrm{2}}\:{or}\:\mathrm{2}{x}=−\mathrm{3}{y} \\ $$$${r}_{\mathrm{2},\mathrm{3}} \:=\:−\mathrm{3}\pm\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$ | ||
Commented by nimnim last updated on 23/Nov/20 | ||
$${Thank}\:{you}\:{Sir}. \\ $$ | ||
Answered by mindispower last updated on 23/Nov/20 | ||
$$\Leftrightarrow{multiplie}\:\left(\mathrm{1}\right).{i\begin{cases}{\mathrm{36}{ix}^{\mathrm{2}} {y}−\mathrm{27}{iy}^{\mathrm{3}} =\mathrm{8}{i}..\left(\mathrm{1}\right)}\\{\mathrm{8}{x}^{\mathrm{3}} −\mathrm{54}{xy}^{\mathrm{2}} =\mathrm{8}....\left(\mathrm{2}\right)}\end{cases}} \\ $$$${and}\:{Eq}\:\mathrm{2}\:\ast\mathrm{2} \\ $$$$\left(\mathrm{1}\right)+\left(\mathrm{2}\right)\Rightarrow−\mathrm{27}{iy}^{\mathrm{3}} +\mathrm{8}{x}^{\mathrm{3}} −\mathrm{54}{xy}^{\mathrm{2}} +\mathrm{36}{ix}^{\mathrm{2}} {y}=\mathrm{16} \\ $$$$\Leftrightarrow\left(\mathrm{3}{iy}\right)^{\mathrm{3}} +\left(\mathrm{2}{x}\right)^{\mathrm{3}} +\mathrm{3}\left(\mathrm{3}{iy}\right)^{\mathrm{2}} \mathrm{2}+\mathrm{3}\left(\mathrm{3}{iy}\right)\left(\mathrm{2}{x}\right)^{\mathrm{2}} =\mathrm{8}\left({i}+\mathrm{1}\right) \\ $$$$\Leftrightarrow\left(\mathrm{3}{iy}+\mathrm{2}{x}\right)^{\mathrm{3}} =\mathrm{8}\sqrt{\mathrm{2}}{e}^{\frac{{i}\pi}{\mathrm{4}}+{i}\mathrm{2}{k}\pi} ....{x},{y}\in\mathbb{R} \\ $$$${this}\:{is}\:{suffisant}\:{to}\:{solve}\:{it} \\ $$$${if}\:{x},{y}\in\mathbb{C} \\ $$$$\left(\mathrm{1}\right)\ast−{i}+...\left(\mathrm{2}\right)\ast\mathrm{2} \\ $$$$\Rightarrow\left(−\mathrm{3}{iy}\right)^{\mathrm{3}} +\mathrm{3}\left(−\mathrm{3}{iy}\right)\left(\mathrm{2}{x}\right)^{\mathrm{2}} +\left(\mathrm{2}{x}\right)^{\mathrm{3}} +\mathrm{3}\left(−\mathrm{3}{iy}\right)^{\mathrm{2}} .\mathrm{2}{x}=−\mathrm{8}{i}+\mathrm{8} \\ $$$$\Rightarrow\left(\mathrm{2}{x}−\mathrm{3}{iy}\right)^{\mathrm{3}} =\mathrm{8}\left(\mathrm{1}−{i}\right)=\mathrm{8}\sqrt{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{4}}+\mathrm{2}{k}_{\mathrm{1}} \pi} ,{k}_{\mathrm{1}} \in\left\{\mathrm{0},\mathrm{1},\mathrm{2}\right\} \\ $$$$\Leftrightarrow\begin{cases}{\mathrm{2}{x}−\mathrm{3}{iy}=\mathrm{8}\sqrt{\mathrm{2}}{e}^{−{i}\frac{\pi}{\mathrm{12}}+\mathrm{2}\frac{{ik}\pi}{\mathrm{3}}} }\\{\mathrm{2}{x}+\mathrm{3}{iy}=\mathrm{8}\sqrt{\mathrm{2}}{e}^{{i}\frac{\pi}{\mathrm{12}}+\mathrm{2}\frac{{k}_{\mathrm{1}} \pi}{\mathrm{3}}} }\end{cases} \\ $$$${solve}\:{and}\:{theck}\:{evry}\:{solution}\:{because}\:{We} \\ $$$${used}\:\Rightarrow \\ $$$$ \\ $$$$ \\ $$$$ \\ $$ | ||
Commented by nimnim last updated on 23/Nov/20 | ||
$${Thank}\:{you}\:{Sir}. \\ $$ | ||
Answered by MJS_new last updated on 23/Nov/20 | ||
$$\mathrm{the}\:\mathrm{easy}\:\mathrm{path} \\ $$$$\mathrm{36}{x}^{\mathrm{2}} {y}−\mathrm{27}{y}^{\mathrm{3}} =\mathrm{8} \\ $$$$\mathrm{4}{x}^{\mathrm{3}} −\mathrm{27}{xy}^{\mathrm{2}} =\mathrm{4} \\ $$$$\mathrm{let}\:{y}={px} \\ $$$$\mathrm{36}{px}^{\mathrm{3}} −\mathrm{27}{p}^{\mathrm{3}} {x}^{\mathrm{3}} =\mathrm{8}\:\Leftrightarrow\:{x}^{\mathrm{3}} =\frac{\mathrm{8}}{\mathrm{36}{p}−\mathrm{27}{p}^{\mathrm{3}} } \\ $$$$\mathrm{4}{x}^{\mathrm{3}} −\mathrm{27}{p}^{\mathrm{2}} {x}^{\mathrm{3}} =\mathrm{4}\:\Leftrightarrow\:{x}^{\mathrm{3}} =\frac{\mathrm{4}}{\mathrm{4}−\mathrm{27}{p}^{\mathrm{2}} } \\ $$$$\Rightarrow \\ $$$$\mathrm{36}{p}−\mathrm{27}{p}^{\mathrm{3}} =\mathrm{8}−\mathrm{54}{p}^{\mathrm{2}} \\ $$$${p}^{\mathrm{3}} −\mathrm{2}{p}^{\mathrm{2}} −\frac{\mathrm{4}}{\mathrm{3}}{p}+\frac{\mathrm{8}}{\mathrm{27}}=\mathrm{0} \\ $$$${p}_{\mathrm{1}} =−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${p}_{\mathrm{2}} =\frac{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}} \\ $$$${p}_{\mathrm{3}} =\frac{\mathrm{4}+\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}} \\ $$$$\Rightarrow \\ $$$${x}_{\mathrm{1}} =−\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{2}}\wedge{y}_{\mathrm{1}} =\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{3}} \\ $$$${x}_{\mathrm{2}} =\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{4}}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)\wedge{y}_{\mathrm{2}} =\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{6}}\left(−\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$${x}_{\mathrm{3}} =−\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{4}}\left(−\mathrm{1}+\sqrt{\mathrm{3}}\right)\wedge{y}_{\mathrm{3}} =−\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{6}}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$ | ||