ROTATION OF AXES (CHANGE OF DIRECTION)

1. Definition: If the axes are rotated through an angle in the same plane by keeping the origin constant, then the transformation is called Rotation of axes. 2. Theorem: To find the co-ordinates of a point ( x, y ) are transformed to plane. Proof: Let x1Ox, yOY 1 are the original axes Let P ( x, y ) be the co-ordinates of the point in the above axes. After rotating the axes through an angle , then the co-ordinates of P be ( X , Y ) in the new axes X 1OX and YOY 1 as in figure.

( X , Y ) when the axes are rotated through an angle about the origin in the same

( X ,Y ) P ( x, y )

N M X

x1

O
1 y1 Y

Since is the angle of rotation, then xOX = yOY = as in the figure.

Since L, M is projections of P on Ox and OX respectively. We can see that

LPM = xOX =
Let N be the projection to PL from M Now x = OL = OQ LQ = OQ NM = OM cos PM sin = X cos Y sin

y = PL = PN + NL = PN + MQ

PM cos + OM sin
= Y cos + X sin

x = X cos Y sin and

y = Y cos + X sin ------ (1)
Solving the above equations to get X and Y, then X = x cos + y sin and

Y = x sin + y cos ---- (2)

From (1) and (2) we can tabulate X x y Note: (i) If the axes are turned through an angle , then the equation of a curve Y

cos

sin
cos

sin

f ( x, y ) = 0 is transformed to f ( X cos Y sin , X sin + Y cos ) = 0

(ii) If f ( X , Y ) = 0 is the transformed equation of a curve when the axes are rotated through an angle then the original equation of the curve is

f ( x cos + y sin , x sin + y cos ) = 0

Theorem: To find the angle of rotation of the axes to eliminate xy term in the equation

ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0
Proof: ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0 Since the axes are rotated through an angle , then x = X cos Y sin ,

y = X sin + Y cos

Now the transformed equation is

a ( X cos Y sin ) + 2h ( X cos Y sin )( X sin + Y cos )
2

+b ( X sin + Y cos ) + 2 g ( X cos Y sin ) + 2 f ( X sin + Y cos ) + c = 0

2

a X 2 cos 2 + Y 2 sin 2 2 XY cos sin +

2h X 2 cos sin + XY cos 2 sin 2 Y 2 sin cos

+b X 2 sin 2 + Y 2 cos 2 + 2 XY cos sin

+2 g ( X cos Y sin ) + 2 f ( X sin + Y cos ) + c = 0

It is in the form
AX 2 + 2 XY a cos sin + h cos 2 sin 2 + b cos sin +

BY 2 + 2GX + 2 FY + C = 0
Since XY term is to be eliminated

( b a ) cos sin + h ( cos 2 sin 2 ) = 0

2h cos 21 = 2 ( a b ) sin cos1 = ( a b ) sin 2

tan 2 =

sin 2 2h = cos 2 a b

1 2h Angle of rotation ( ) = Tan 1 2 a b

Note: The angle of rotation of the axes to eliminate xy term in

ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0 is

PROBLEMS 1. When the axes are rotated through an angle 30 , find the new co-ordinates of the following points. i) (0, 5) Sol. i) Given = 30 Old co-ordinates are (0,5) i.e., x=0, y = 5 ii) (-2, 4) iii) (0, 0)

x1 = x cos + y sin
= 0.cos30 + 5.sin 30 =

5 2

y1 = x sin + y cos
0.sin 30 + 5.cos30 = 5 3 2

New co-ordinates are ,

5 5 3 2 2

ii) Old co-ordinates are (-2,4) x=2, y=4

x1 = x cos + y sin

( 2 ) .cos30 + 4.sin 30
2. 3 1 + 4. = 3 + 2 2 2

y1 = x sin + y cos ( 2 ) sin 30 + 4cos30

= 2. + 4. = 1+ 2 3

1 2

3 2

New co-ordinates are

3 + 2, 1 + 2 3

iii) Given ( x, y ) = ( 0,0 ) and = 30

x = ( 0, y ) x = x.cos30 y sin 30
= 0.

3 1 0. = 0 2 2

y = x.sin 30 + y.cos30
1 3 0. + 0. =0 2 2
New co-ordinates of the point are (0, 0) 2. When the axes are rotated through an angle 60 , the new co-ordinates of three points are the following i) (3, 4) ii) (-7, 2) iii) (2, 0) Find their original co-ordinates

Sol. i) Given = 60 New co-ordinates are (3, 4)

x1 = 3, y1 = 4 x = x1 cos y1 sin
3.cos 60 4.sin 60
= 3.

1 2

4. 3 3 4 3 = 2 2

y1 = x1 sin + y1 cos
= 3sin 60 + 4.cos 60

3.

3 1 4+ 3 + 4. = 2 2 2

Co-ordinates of P are

3 4 3 4+3 3 , 2 2

ii) New coordinates are (-7,2)

x1 = 7, y1 = 2 x = x1 cos y1 sin
= ( 7 ) cos 60 2.sin 60 = 7. 2.

1 2

3 7 2 3 = 2 2

y = x1 sin + y1.cos
= 7.sin 60 + 2.cos 60 = 7.

3 1 27 3 + 2. = 2 2 2

Co-ordinates of Q are

7 2 3 2 7 3 , 2 2

iii) New co-ordinates are (2, 0)

x1 = 2, y1 = 0 x = x1 cos y1 sin
= 2.cos 60 0.sin 60 = 2. 0.

1 2

3 = 1 0 = 1 2

y = x1 sin + y1 cos
= 2.sin 60 + 0.cos 60

= 2.

3 1 + 0. = 3 2 2

Co-ordinates of R are 1, 3 3.

)
?

Find the angle through which the axes are to be rotated so as to remove the xy term in the equation. x 2 + 4 xy + y 2 2 x + 2 y 6 = 0

Sol. Comparing the equation

x 2 + 4 xy + y 2 2 x + 2 y 6 = 0 with ax 2 + 2hxy + by 2 + 2 gx + 2 fy + c = 0
a = 1, h =2, b=1, g=-1, f=1, c=-6 Let be the angle of rotation of axes, then

= tan 1 2 a b
=

2h

1 4 1 1 4 tan 1 = tan 2 1 1 2 0 1 1 tan 1 ( ) = 2 2 2

4
SHORT ANSWERS QUESTIONS

1.

When the axes are rotated through an angle 45 , the transformed equation of a curve is 17 x 2 16 xy + 17 y 2 = 225 .Find the original equation of the curve.?

Sol. Angle of rotation = = 45

x1 = x cos + y sin = x cos 45 + y sin 45

=

x+ y 1 y = x sin + y cos = x sin 45 + y cos 45 2

x + y 2
The original equation of

17 x1 16 x1 y1 + 17 y1 = 225 is
= 17 x + y 16 x + y x + y + 17 x + y = 225 2 2 2 2

2 2

= 17 (

x 2 + y 2 + 2 xy 2

) 16 ( y

x2 2

) + 17 ( x

+ y 2 2 xy 2

) = 225

=17 x 2 + 17 y 2 + 345 xy 16 y 2 + 16 x 2 +17 x 2 + 17 y 2 34 xy = 450 = 50 x 2 + 18 y 2 = 450

x 2 + y 2 = 9 is the original equation

2. when the axes are rotated through an angle , find the transformed equation of x cos + y sin = p ?

Sol. The given equation is x cos + y sin = p The axes are rotated through an angle

x = x1 cos y1 sin y = x1 sin + y1 cos

The given equation transformed to

( x cos y sin ) cos + ( x sin + y cos ) sin = p

1 1 1 1

x1 cos 2 + sin 2 = p x1 = p
The equation transformed to x = p

3.

When the axes are rotated through an angle 6 . Find the transformed equation of x 2 + 2 3 xy y 2 = 2a 2

Sol. Since =

, x = X cos Y sin Y sin

X = X cos

X.

3 1 3X Y Y. = 2 2 2

y = X sin + Y cos = X .sin

= X . + Y.

+ Y cos

1 2

3 X + 3Y = 2 2

Transformed equation is
3 X Y X + 3Y f , =0 2 2

3X Y 3 X Y X + 3Y X + 3Y 2 = 2a + 2 3 2 2 2 2
2
2 2 3x 2 2 3 + Y 2 2 3 3 X XY + 3 XY 3Y = + 4 4

X 2 + 3Y 2 + 2 3 XY = 2a 2 4

3 X 2 2 3 XY + Y 2 + 2 3 3 X 2 + 2 XY + 3Y 2

X 2 + 3Y 2 + 3 XY = 8a 2
3 X 2 2 3 + Y 2 + 6 X 2 + 4 3 XY 6Y 2 X 2 3Y 2 2 3 XY = 8a 2

8 X 2 8Y 2 = 8a 2 X 2 Y 2 = a 2

4.

When the axes are rotated through an angle of 3 x 2 + 10 xy + 3 y 2 = 9 ?

, find the transformed equation

Sol. Given equation is

3x 2 + 19 xy + 3 y 9 = 0 ........... (1)
Angle of rotation of axes = =

Let (X,Y) be the new co-ordinates of ( x. y )

x = X cos Y sin

X cos

y sin

X Y 2

4
+ Y cos

y = X sin + Y cos = X sin

2

X + Y Transformed equation of (1) is 2

2

(X 3

X Y X Y X + Y X + Y 2 + 10 + 3 9 = 0 2 2 2 2
2

2 XY + Y 2 2

) + 10 ( X

Y2 2

) +3 ( X

+ 2 XY + Y 2 2

) 9 = 0

= 3 X 2 6 XY + 3Y 2 + 10 X 2 10Y 2 +3 X 2 + 6 XY + 3Y 2 18 = 0

16 X 2 4Y 2 18 = 0 8 X 2 2Y 2 = 9 8 X 2 2Y 2 = 9
5. Find the transformed equation of17 x 2 16 xy + 17 y 2 = 225 when the axes are rotated through an angle 45 ?

Sol. Let ( x, y ) the original equation of ( X , Y ) Angle of rotation = 45 Now X = x cos y sin

= x cos 45 y sin 45 =

x y 2

Y = x sin + y cos
= x sin 45 + y cos 45 =

x+ y 2

The transformed equation is 45

x y x+ y , f =0 2 2
x y x y X + Y 17 16 2 2 2
2

X +Y +17 = 225 2

x 2 + y 2 2 xy x2 y 2 x 2 + y 2 + 2 xy 17 16 +17 = 225 2 2 2
2 17 X 2 + 17Y 2 34 XY 16 X 2 + 16Y 2 + 17 X 2 +17Y + 34 XY = 450

18 X 2 + 50Y 2 = 450

9 X 2 + 25Y 2 = 225

GENERAL TRANSFORMATIONS
1. Definition: If the axes are rotated through an angle after shifting the origin in the same plane, then the transformation is called General Transformation New origin A = ( x1 , y1 ) , angle of rotation = as in figure

Y N
A

( X ,Y ) P ( x, y )
X M

X1
O

Y1
L

y1
We get the transformed equations as

x = x1 + X cos Y sin y = y1 + X sin + Y cos

X = ( x x1 ) cos + ( y y1 ) sin
Y = ( x x1 ) sin + ( y y1 ) cos
We can easily understand the translation and rotation satisfy commutative property.

PROBLEMS. 1. When the origin is shifted to (-2,-3) and the axes are rotated through an angle 45 find the transformed of 2x2 + 4xy 5 y2 + 20x 22 y 14 = 0 ?

Sol. Here ( h, k ) = ( 2, 3) , h = 2, k = 3

= 45
Let x1 , y1 be the new co-ordinates of any point ( x, y ) is the plane after transformation
x = x1 cos y1 sin + h = 2 x + x1 cos 45 y1 sin 45

= 2 +

x1 y1 2

y = x1 sin + y1 cos + k = x1 sin 45 + y1 cos 45 3

x1 + y1 3 + 2
The transformed equation is

x1 y1 x1 y1 x1 + y1 2 2 + 4 2 3 2 2 2
x1 + y1 x1 y1 x1 + y1 5 3 + 20 2 22 3 14 = 0 2 2 2
2

x1 + y1 2 2

+ 42 2 x y + 4

x12 y12 x1 y1 x1 + y1 3 2 + 6 2 2 2

) (

x1 + y1 5 2

+ 9 3 2 x + y + 10 2 x1 y1 2 2 11 2

x1 + y1 3 2 14 = 0

(x

+ y1
2

+ 8 4 2 x1 y1
2

)
)
2

+2 x1 y1 6 2 x1 y1

4 2 x1 + y1 + 24 = 0 5 ( x1 + y1 ) 45 + 15 2 ( x1 + y1 ) + 10 2 ( x1 y1 ) 40 11 2 x1 + y1 + 66 14 = 0

x1 + y1 2 x1 y1 + 2 x1 2 y1

5 2

(x

12

+ y1 + 2 x1 y1 1 = 0

1 12 7 12 x y 7 x1 y1 1 = 0 2 2
i.e, x1 7 y1 14 x1 y1 2 = 0 The transformed equation is (dropping dashes)
2 2

x 2 7 y 2 14 xy 2 = 0

PROBLEMS FOR PRACTICE 1. Find the transformed equation of 5 x 2 + 4 xy + 8 y 2 12 x 12 y = 0 . When the origin is shifted to 1, by translation of axes. 2. When the origin is shifted to (3,-4) by the translation of axis and the transformed equation is x 2 + y 2 = 4 , find the original equation. 3. When the origin is shifted to (2,3) by the translocation of axes, the co-ordinates of a point p are changed as (4,-3). Find the co-ordinates of P in the original system. Find the point to which the origin is to be shifted by the translation of axes so as to remove the first degree terms from the equation

1 2

4.

ax 2 + by 2 + 2 gx + 2 fy + c = 0 , where a 0, b 0
5. If the point P changes to (4,-3) when the axes are rotated through an angle of 135 , find the coordinates of P with respect to the original system. Show that the axes are to be rotated through an angle of

6.

1 2h Tan 1 so as 2 a b

to remove the xy term from the equation ax 2 + 2hxy + by 2 = 0 , if a b and through the angle

, if a = b