Word Count : 264

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यह शांकव $$ x^2-2xy+y^2-3x+2y+3=0 $$ का अनुरेखण (tracing) करें। ## 1) रूप का निर्धारण यहाँ $A=1,\;B=-2,\;C=1$. विभेदक $B^2-4AC=4-4=0\Rightarrow$ यह परवलय (parabola) है, परंतु $xy$ पद के कारण घुमी हुई है। ## 2) $xy$ पद हटाने हेतु घूर्णन $$ \tan 2\theta=\frac{B}{A-C}=\frac{-2}{0}\Rightarrow 2\theta=90^\circ\;\Rightarrow\;\theta=45^\circ . $$ घूर्णित निर्देशांक $(X,Y)$ लें: $$ x=\frac{X-Y}{\sqrt2},\qquad y=\frac{X+Y}{\sqrt2}. $$ ध्यान दें $x^2-2xy+y^2=(x-y)^2$. ## 3) घूर्णन के बाद समीकরণ $$ (x-y)^2-3x+2y+3=0 $$ $$ \Rightarrow 2Y^2-\frac{1}{\sqrt2}X+\frac{5}{\sqrt2}Y+3=0 _________ _____ ________ _____ _________ _____ ________ ________ _________.
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