Analytical geometry of two dimensions by R.M. Khan

1.👉 Transformation of axes

The coordinates of a point depend on the position of axes. Thus the coordinates of a point and consequently the equation of a locus will be changed with the alteration of origin without the alteration of direction axes, or by altering the direction of axes and keeping the origin fixed, or by altering the origin and also the direction of axes. Either of these processes is known as transformation of ares or transformation of Coordinates. 

 2.👉 Change of origin without change of direction of axes

• Let (x, y) be the coordinates of P w.r.t rectangular axes OX and OY and (x',y') be the coordinates of it w.r.t. a new set of axes O'X' and O'Y' which are parallel to the original axes OX and OY respectively. 
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• Let (α,β) be the coordinates of the new origin O' w.r.t. axes OX and OY PN is perpendicular to OX and it meets O'X' at N'. O'T is perpendicular to OX.

ON = x,.   NP = y 

ƠN' =x', N P =y'

ΟΤ=α  ; ΤΟ'=β         

Now

x=ON = OT +TN = OT +O'N' = α+x',

 y=NP=NN'+ N'P=TO'+N'P= β+y'

3.👉 Rotation of rectangular axes in their own plane without changing the origin

Let the original axes OX and OY be rotated through an angle θ in the anti-clockwise direction. In the adjoining figure OX' and OY' are the new set of axes. Let (x, y)

and (x', y') be the coordinates of the same point P referred to OX, OY and OX' ,OY' respectively.

PM and PM' are perpendicular to OX and OX' respectively. PO is Joined. Here angleXOX'=θ

Let angleΧ'ΟΡ=α

From the figure

Now

χ=OM = OP co(θ+ α)

=OP cosα cosθ -OP sinα sinθ

=OM' cosθ - M'P sinθ

=x' cosθ-y'sinθ

y=MP = OP sin(θ+α)
=OP cosα sinθ+ sinα cosθ
=OM' sinθ + M'P cosθ
=x'sinθ+ y' cosθ

Hence the change from (z, y) to (z. v') is given by

x=x'cosθ - y' sinθ and
y=x'sinθ + y' cosθ

4.👉 Combination of translation and rotation

If the origin O of a set of rectangular axes (OX, OY) is shifted to O (α,β)  (referred to OX and OY )without changing the direction of axes and then the axes are rotated through an angle 0 in the anti-clockwise direction, the total effective changes in the coordinates (x, y) of a point are given by

x= α  + x'' cosθ-y''sinθ

 y=β  +x''sinθ + y'' cosθ

(x",y'') are the coordinates of the point referred to the final set of axes.

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How to download R.M. Khan exercise solutions

Exercise (iii) /page no 27

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