# Current Affairs JEE Main & Advanced

## Transformation of Axes

Category : JEE Main & Advanced

(1) Shifting of origin without rotation of axes : Let $P\equiv (x,y)$with respect to axes $OX$ and $OY$.

Let $O'\equiv (\alpha ,\beta )$ with respect to axes $OX$ and $OY$ and let $P\equiv (x',y')$ with respect to axes $O'X'$  and  $O'Y',$  where $OX$ and  $O'X'$  are parallel and $OY$ and  $O'Y'$ are parallel.

Then $x=x'+\alpha ,\text{ }y=y'\,+\beta$

or  $x'=x-\alpha ,\text{ }y'=y-\beta$

Thus if origin is shifted to point $(\alpha ,\beta )$ without rotation of axes, then new equation of curve can be obtained by putting $x+\alpha$ in place of $x$ and $y+\beta$ in place of  $y$.

(2) Rotation of axes without changing the origin : Let  $O$ be the origin. Let $P\equiv (x,y)$ with respect to axes $OX$ and $OY$ and let $P\equiv (x',y')$ with respect to axes $OX'$ and $OY'$ where $\angle X'OX=\angle YOY'=\theta$

then      $x=x'\cos \theta -y'\sin \theta$

$y=x'\sin \theta +y'\cos \theta$

and      $x'=x\cos \theta +y\sin \theta$

$y'=-x\sin \theta +y\cos \theta$

The above relation between $(x,y)$ and $(x',y')$ can be easily obtained with the help of following table

 $x\downarrow$ $y\downarrow$ $x'\to$ $y'\to$ $\cos \theta$ $-\sin \theta$ $\sin \theta$ $\cos \theta$

(3) Change of origin and rotation of axes : If origin is changed to $O'(\alpha ,\beta )$ and axes are rotated about the new origin $O'$ by an angle $\theta$ in the anti-clockwise sense such that the new co-ordinates of $P(x,y)$ become $(x',y')$ then the equations of transformation will be $x=\alpha +x'\cos \theta -y'\sin \theta$ and  $y=\beta +x'\sin \theta +y'\cos \theta$

(4) Reflection (Image of a point) : Let $(x,y)$be any point, then its image with respect to

(i) x-axis $\Rightarrow$ $(x,-y)$

(ii) y-axis $\Rightarrow$ $(-x,y)$

(iii) origin $\Rightarrow$ $(-x,-y)$

(iv) line $y=x$$\Rightarrow$$(y,x)$

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