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Question Number 136607 by EDWIN88 last updated on 23/Mar/21
What is the value of a^2 +b^2  if ax+by=3  bx−ay = 4 and x^2 +y^2  = 4
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \:\mathrm{if}\:\mathrm{ax}+\mathrm{by}=\mathrm{3} \\ $$$$\mathrm{bx}−\mathrm{ay}\:=\:\mathrm{4}\:\mathrm{and}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$
Answered by MJS_new last updated on 24/Mar/21
 { ((ax+by=3)),((bx−ay=4)) :} ⇔  { ((a=((3x−4y)/(x^2 +y^2 )))),((b=((4x+3y)/(x^2 +y^2 )))) :} ⇒ a^2 +b^2 =((25)/(x^2 +y^2 ))=((25)/4)
$$\begin{cases}{{ax}+{by}=\mathrm{3}}\\{{bx}−{ay}=\mathrm{4}}\end{cases}\:\Leftrightarrow\:\begin{cases}{{a}=\frac{\mathrm{3}{x}−\mathrm{4}{y}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }}\\{{b}=\frac{\mathrm{4}{x}+\mathrm{3}{y}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }}\end{cases}\:\Rightarrow\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\frac{\mathrm{25}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\frac{\mathrm{25}}{\mathrm{4}} \\ $$
Answered by mindispower last updated on 24/Mar/21
(ax+by)^2 +(bx−ay)^2 =(a^2 +b^2 )(x^2 +y^2 )...
$$\left({ax}+{by}\right)^{\mathrm{2}} +\left({bx}−{ay}\right)^{\mathrm{2}} =\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)… \\ $$$$ \\ $$

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