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Question Number 38879 by Rio Mike last updated on 30/Jun/18
Find the equation of the line through  (2,−3) which make angles 45° with  the line 2x − y = 2.
$${Find}\:{the}\:{equation}\:{of}\:{the}\:{line}\:{through} \\ $$$$\left(\mathrm{2},−\mathrm{3}\right)\:{which}\:{make}\:{angles}\:\mathrm{45}°\:{with} \\ $$$${the}\:{line}\:\mathrm{2}{x}\:−\:{y}\:=\:\mathrm{2}. \\ $$
Answered by MrW3 last updated on 30/Jun/18
line 2x−y=2:  tan θ=m=2  k=tan (θ±(π/4))=((tan θ±tan (π/4))/(1∓tan θ tan (π/4)))=((m±1)/(1∓m))  =((2±1)/(1∓2))= { ((−3)),((1/3)) :}  ⇒eqn. of new line:  ((y+3)/(x−2))=−3 or (1/3)  ⇒y+3=−3(x−2)  ⇒3x+y=3  or  ⇒3(y+3)=x−2  ⇒x−3y=11
$${line}\:\mathrm{2}{x}−{y}=\mathrm{2}: \\ $$$$\mathrm{tan}\:\theta={m}=\mathrm{2} \\ $$$${k}=\mathrm{tan}\:\left(\theta\pm\frac{\pi}{\mathrm{4}}\right)=\frac{\mathrm{tan}\:\theta\pm\mathrm{tan}\:\frac{\pi}{\mathrm{4}}}{\mathrm{1}\mp\mathrm{tan}\:\theta\:\mathrm{tan}\:\frac{\pi}{\mathrm{4}}}=\frac{{m}\pm\mathrm{1}}{\mathrm{1}\mp{m}} \\ $$$$=\frac{\mathrm{2}\pm\mathrm{1}}{\mathrm{1}\mp\mathrm{2}}=\begin{cases}{−\mathrm{3}}\\{\frac{\mathrm{1}}{\mathrm{3}}}\end{cases} \\ $$$$\Rightarrow{eqn}.\:{of}\:{new}\:{line}: \\ $$$$\frac{{y}+\mathrm{3}}{{x}−\mathrm{2}}=−\mathrm{3}\:{or}\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow{y}+\mathrm{3}=−\mathrm{3}\left({x}−\mathrm{2}\right) \\ $$$$\Rightarrow\mathrm{3}{x}+{y}=\mathrm{3} \\ $$$${or} \\ $$$$\Rightarrow\mathrm{3}\left({y}+\mathrm{3}\right)={x}−\mathrm{2} \\ $$$$\Rightarrow{x}−\mathrm{3}{y}=\mathrm{11} \\ $$

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