TRANSFORMATION OF RECTANGULAR AXES

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Types of the transformation of axes

There are three distinct types of transformation of co-ordinates, namely

 (i) translation 

(ii) rotation 

(iii) translation and rotation.

These are called orthogonal transformation when both systems of axes are rectangular. The combination of transformation and rotation is called is a rigid body motion. 

The Let (x, y) be the coordinates of a point in the old system, (x′,y′) the coordinates of the same point in the new system,(α,β) the coordinates of the new origin with respect to the old system and θ the angle of rotation. The formulae for the coordinate transformation due to a (1) (translation): r=x'+a, y=y' + B. translation, rotation, translation and rotation are  as .

 (1) translation -   x=x′+α   y=y′+β

 (2) rotation-   X =X′ cosΘ- Y′sinΘ ;       Y= X′ sinΘ+ Y′ cosΘ

         

 (3) translation and rotation-  X =X′ cosΘ- Y′sinΘ + α  ;     Y= X′ sinΘ+ Y′ cosΘ+β

Example:1

 Find the equation of the line       when the origin is shifted to the point (a,b)

solution:

since the origin is shifted to the point (a,b) , then x = X+a and y =Y+b

The given equation is     . Then

     

or,           

or,            

                   

This is the transformed form of the given line.

Example-2
 

Transform the equation y²-2y=x with respect to parallel axes through (-1,1)           

solution: 

since the origin is shifted to the point (-1,1) then     x =X-1 and y=Y+1

The given equation is  y²-2y=x , so,

(Y+1)²-2(Y+1) = X-1

or,  (Y+1)(Y-1) = X-1

or, Y²-1 = X-1

or. Y² =X

This is the transformed form of the given equation

Example-3

If under rotation an expression of the form ax+by changes AX+BY. Show that a²+b² remains invariant. 

Solution:

Let be the angle of rotation . Then 

x =X cosΘ- YsinΘ ; y= X sinΘ+ Y cosΘ

ax+by =a(X cosΘ -Y sinΘ) + b (X sinΘ + Y cosΘ)

              =   (a cosΘ + b sinΘ)X + (-a sinΘ + b cosΘ)Y

               

              =   AX +BY (given)

Therefore,    A = acosΘ +bsinΘ   ..........................1

and               B = -asinΘ+bcosΘ   .......................2

Squaring (1)  and (2), we get

                          

             A²+B² = (acosΘ +bsinΘ)² +  (-asinΘ+bcosΘ)²

                       = a²(cos²Θ+sin²Θ)+b²(cos²Θ + sin²Θ)

                       =  a²+b²

Thus  a²+b² remains  invariant .

Example-4

Show that the equation x²+y²=a² is an invariant under rotation of axes.

solution :

x =X cosΘ- YsinΘ ;    y= X sinΘ+ Y cosΘ

we have ,

                                x ²+y²=a²

                              or,  (X cosΘ- YsinΘ)² + (X sinΘ+ Y cosΘ)²=a²

                               or,  X²(cos²Θ+sin²Θ)+Y²(cos²Θ + sin²Θ)=a²

                                or,  X ²+Y²=a²

                                

                    Thus the equation x ²+y²=a²  is an invariant under rotation of axes.