Menu Close

x-y-k-x-3y-6-x-2-y-2-min-

Tinku Tara 0 Comments
Pin




Question Number 127088 by MathSh last updated on 26/Dec/20
{ ((x+y=k)),((x−3y=6)) :} (x^2 −y^2 )_(min) =?
$$\begin{cases}{{x}+{y}={k}}\\{{x}−\mathrm{3}{y}=\mathrm{6}}\end{cases} \\ $$$$\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)_{{min}} =? \\ $$
Commented by hknkrc46 last updated on 26/Dec/20
✓ x − 3y = (x + y) − 4y = k − 4y = 6 ⇒ 4y = k − 6 ⇒ y = ((k − 6)/4) ✓ x + y = x + ((k − 6)/4) = k ⇒ x = k − ((k − 6)/4) = ((3k + 6)/4) ✓ x^2 − y^2 = (((3k + 6)/4))^2 − (((k − 6)/4))^2 = ((k^2 + 6k)/2) = (((k + 3)^2 − 9)/2) (min : k = −3) = − (9/2)
$$\checkmark\:\boldsymbol{{x}}\:−\:\mathrm{3}\boldsymbol{{y}}\:=\:\left(\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\right)\:−\:\mathrm{4}\boldsymbol{{y}}\:=\:\boldsymbol{{k}}\:−\:\mathrm{4}\boldsymbol{{y}}\:=\:\mathrm{6} \\ $$$$\Rightarrow\:\mathrm{4}\boldsymbol{{y}}\:=\:\boldsymbol{{k}}\:−\:\mathrm{6}\:\Rightarrow\:\boldsymbol{{y}}\:=\:\frac{\boldsymbol{{k}}\:−\:\mathrm{6}}{\mathrm{4}} \\ $$$$\checkmark\:\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:=\:\boldsymbol{{x}}\:+\:\frac{\boldsymbol{{k}}\:−\:\mathrm{6}}{\mathrm{4}}\:=\:\boldsymbol{{k}} \\ $$$$\Rightarrow\:\boldsymbol{{x}}\:=\:\boldsymbol{{k}}\:−\:\frac{\boldsymbol{{k}}\:−\:\mathrm{6}}{\mathrm{4}}\:=\:\frac{\mathrm{3}\boldsymbol{{k}}\:+\:\mathrm{6}}{\mathrm{4}} \\ $$$$\checkmark\:\boldsymbol{{x}}^{\mathrm{2}} \:−\:\boldsymbol{{y}}^{\mathrm{2}} \:=\:\left(\frac{\mathrm{3}\boldsymbol{{k}}\:+\:\mathrm{6}}{\mathrm{4}}\right)^{\mathrm{2}} −\:\left(\frac{\boldsymbol{{k}}\:−\:\mathrm{6}}{\mathrm{4}}\right)^{\mathrm{2}} =\:\frac{\boldsymbol{{k}}^{\mathrm{2}} \:+\:\mathrm{6}\boldsymbol{{k}}}{\mathrm{2}} \\ $$$$=\:\frac{\left(\boldsymbol{{k}}\:+\:\mathrm{3}\right)^{\mathrm{2}} \:−\:\mathrm{9}}{\mathrm{2}}\:\left(\boldsymbol{{min}}\::\:\boldsymbol{{k}}\:=\:−\mathrm{3}\right)\:=\:−\:\frac{\mathrm{9}}{\mathrm{2}} \\ $$
Commented by MathSh last updated on 27/Dec/20
Thanks sir
$${Thanks}\:{sir} \\ $$
Answered by Olaf last updated on 26/Dec/20
y = ((k−6)/4), x = ((3(k+2))/4) x^2 −y^2 = (9/(16))(k+2)^2 −(1/(16))(k−6)^2 x^2 −y^2 = (9/(16))(k^2 +4k+4)−(1/(16))(k^2 −12k+36) x^2 −y^2 = (1/2)k^2 +3k x^2 −y^2 = (k/2)(k+6) If k = −3, (x^2 −y^2 )_(min) = −(9/2) x = −(3/4) y = −(9/4)
$$ \\ $$$${y}\:=\:\frac{{k}−\mathrm{6}}{\mathrm{4}},\:{x}\:=\:\frac{\mathrm{3}\left({k}+\mathrm{2}\right)}{\mathrm{4}} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:=\:\frac{\mathrm{9}}{\mathrm{16}}\left({k}+\mathrm{2}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{16}}\left({k}−\mathrm{6}\right)^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:=\:\frac{\mathrm{9}}{\mathrm{16}}\left({k}^{\mathrm{2}} +\mathrm{4}{k}+\mathrm{4}\right)−\frac{\mathrm{1}}{\mathrm{16}}\left({k}^{\mathrm{2}} −\mathrm{12}{k}+\mathrm{36}\right) \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}{k}^{\mathrm{2}} +\mathrm{3}{k} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:=\:\frac{{k}}{\mathrm{2}}\left({k}+\mathrm{6}\right) \\ $$$$\mathrm{If}\:{k}\:=\:−\mathrm{3},\:\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)_{{min}} \:=\:−\frac{\mathrm{9}}{\mathrm{2}} \\ $$$${x}\:\:=\:−\frac{\mathrm{3}}{\mathrm{4}} \\ $$$${y}\:=\:−\frac{\mathrm{9}}{\mathrm{4}} \\ $$
Commented by MathSh last updated on 27/Dec/20
Thanks sir
$${Thanks}\:{sir} \\ $$
Answered by Olaf last updated on 26/Dec/20
sorry sir ! I read x^2 +y^2 instead of x^2 −y^2 . I′m tired !
$${sorry}\:{sir}\:!\:{I}\:{read}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:{instead}\:{of} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} .\:{I}'{m}\:{tired}\:! \\ $$
Commented by MathSh last updated on 27/Dec/20
Yes, thanks sir
$${Yes},\:{thanks}\:{sir} \\ $$
Answered by mathmax by abdo last updated on 26/Dec/20
{ ((x+y=k)),((x−3y=6 ⇒ { ((4y=k−6 ⇒ { ((y=((k−6)/4))),((x=k−((k−6)/4))) :})),((x=k−y)) :})) :} ⇒ { ((x=((3k+6)/4))),((y=((k−6)/4) ⇒ x^2 −y^2 =(1/(16)){(3k+6)^2 −(k−6)^2 }=f(k) ⇒)) :} f(k)=(1/(16))(9k^2 +36k +36−k^2 +12k−36} =(1/(16))(8k^2 +48k) =(k^2 /2) +3k ⇒f^′ (k)=k+3 and f^′ (k)=0 ⇒k=−3 k −∞ −3 +∞ f^′ − 0 + f decr f(−3) incr ⇒ minf(k)=f(−3) =(9/2)−9 =−(9/2) =min(x^2 −y^2 )
$$\begin{cases}{\mathrm{x}+\mathrm{y}=\mathrm{k}}\\{\mathrm{x}−\mathrm{3y}=\mathrm{6}\:\:\:\:\:\:\Rightarrow\:\begin{cases}{\mathrm{4y}=\mathrm{k}−\mathrm{6}\:\:\Rightarrow\begin{cases}{\mathrm{y}=\frac{\mathrm{k}−\mathrm{6}}{\mathrm{4}}}\\{\mathrm{x}=\mathrm{k}−\frac{\mathrm{k}−\mathrm{6}}{\mathrm{4}}}\end{cases}}\\{\mathrm{x}=\mathrm{k}−\mathrm{y}}\end{cases}}\end{cases} \\ $$$$\Rightarrow\begin{cases}{\mathrm{x}=\frac{\mathrm{3k}+\mathrm{6}}{\mathrm{4}}}\\{\mathrm{y}=\frac{\mathrm{k}−\mathrm{6}}{\mathrm{4}}\:\Rightarrow\:\:\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \:=\frac{\mathrm{1}}{\mathrm{16}}\left\{\left(\mathrm{3k}+\mathrm{6}\right)^{\mathrm{2}} −\left(\mathrm{k}−\mathrm{6}\right)^{\mathrm{2}} \right\}=\mathrm{f}\left(\mathrm{k}\right)\:\Rightarrow}\end{cases} \\ $$$$\mathrm{f}\left(\mathrm{k}\right)=\frac{\mathrm{1}}{\mathrm{16}}\left(\mathrm{9k}^{\mathrm{2}} +\mathrm{36k}\:+\mathrm{36}−\mathrm{k}^{\mathrm{2}} +\mathrm{12k}−\mathrm{36}\right\}\:=\frac{\mathrm{1}}{\mathrm{16}}\left(\mathrm{8k}^{\mathrm{2}} +\mathrm{48k}\right) \\ $$$$=\frac{\mathrm{k}^{\mathrm{2}} }{\mathrm{2}}\:+\mathrm{3k}\:\:\Rightarrow\mathrm{f}^{'} \left(\mathrm{k}\right)=\mathrm{k}+\mathrm{3}\:\:\mathrm{and}\:\mathrm{f}^{'} \left(\mathrm{k}\right)=\mathrm{0}\:\Rightarrow\mathrm{k}=−\mathrm{3} \\ $$$$\mathrm{k}\:\:\:\:\:\:\:\:\:\:\:\:−\infty\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\infty \\ $$$$\mathrm{f}^{'} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:+ \\ $$$$\mathrm{f}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{decr}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\left(−\mathrm{3}\right)\:\:\:\mathrm{incr}\:\:\Rightarrow \\ $$$$\mathrm{minf}\left(\mathrm{k}\right)=\mathrm{f}\left(−\mathrm{3}\right)\:=\frac{\mathrm{9}}{\mathrm{2}}−\mathrm{9}\:\:=−\frac{\mathrm{9}}{\mathrm{2}}\:=\mathrm{min}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \right) \\ $$
Commented by MathSh last updated on 27/Dec/20
Thanks sir
$${Thanks}\:{sir} \\ $$

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *