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  • Write the equation  in terms of a rotated <img alt="x^{\prime} y^{\prime}" src="https://learn.careers360.com/latex-

Write the equation x y=1 in terms of a rotated x^{\prime} y^{\prime}  -system if the angle of rotation from the x -axis to the x^{\prime} -axis is  45^{\circ}.

Option: 1

{\frac{x^{\prime 2}}{2}-\frac{y^{\prime 2}}{2}=1}


Option: 2

{\frac{x^{\prime 2}}{2}+\frac{y^{\prime 2}}{2}=1}


Option: 3

{\frac{x^{\prime 2}}{2}-\frac{y^{\prime 2}}{2}=2}


Option: 4

None of these


Answers (1)

best_answer

 

 

Rotation of Axes About Origin -

Rotation of Axes About Origin  

P(x, y) is the point in the original coordinate system and axes are rotated by an angle ? anticlockwise direction about the origin. Then, the coordinates of point P with respect to the new coordinate system is (X, Y) =  (x cos ? + y sin ?, y cos ? - x sin ? ).   

OX and OY are original system of coordinate axes and OX’ and OY’ are the new system of coordinate axes. PM and PN are perpendicular to OX and OX’ and also NL and NQ  perpendicular OX and PM.

We have

From the figure:

\\\text{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;OM = x, PM = y ON = X and PN = Y}\\\text{Now,\;\;\;\;\;\;\;\;\;\;\;\;x = OM = OL - ML}\\\because \text{angle between two lines = angles between their perpendiculars}\\\mathrm{\;\;\;\;\;\;\;\;\;=OL-QN= ON\cos\theta-PN\sin\theta}\\\mathrm{\;\;\;\;\;\;\;\;\;=X\cos\theta-Y\sin\theta}\\\mathrm{i.e.\;\;\mathbf{x=X\cos\theta-Y\sin\theta}\;\;\;\;\;\;\;\;\;\;\ldots(i)}\\\text{And,\;\;\;\;\;\;\;\;\;\;\;\;y = PM = PQ + QM = PQ + NL}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= PN\cos\theta+ON\sin\theta}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=Y\cos\theta+X\sin\theta}\\\mathrm{i.e.\;\;\mathbf{y=Y\cos\theta+X\sin\theta}\;\;\;\;\;\;\;\;\;\;\ldots(ii)}

Solving (i) and (ii), we get

X = x cos ? + y sin ?

Y = y cos ? - x sin ?

AID TO MEMORY

\begin{array}{|c|c|c|}\hline & {x} & {y} \\ \hline X & {\cos \theta} & {\sin \theta} \\ \hline Y & {-\sin \theta} & {\cos \theta} \\ \hline\end{array}

-

 

 

 

x=x^{\prime} \cos 45^{\circ}-y^{\prime} \sin 45^{\circ}\\ y=x^{\prime} \sin 45^{\circ}+y^{\prime} \cos 45^{\circ}\\ x y=1\\ \begin{array}{r}{\left[\frac{1}{\sqrt{2}}\left(x^{\prime}-y^{\prime}\right)\right]\left[\frac{1}{\sqrt{2}}\left(x^{\prime}+y^{\prime}\right)\right]=1} \\ {\frac{1}{2}\left(x^{\prime 2}-y^{\prime 2}\right)=1} \\ {\frac{x^{\prime 2}}{2}-\frac{y^{\prime 2}}{2}=1}\end{array}

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Sayak

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