Title XI - maths - chapter 12 - TRANSFORMATION OF AXES(11.03.2015)-(48-61).pdf ✅

4 3 MAINS ADVANCED TRANSFORMATION OF AXES MATHS BY ROHIT SIR MOB. 8218498992 [email protected] iv) To remove the first degree terms from the equation 2 2 ax 2hxy by 2gx 2fy c 0       the origin is to be shifted to the point  1 1  2 2 , ,            hf bg gh af x y ab h ab h , 2 ab h 0   . In this case, the transformed equation is aX2 + 2hXY + bY2 + (gx1 + fy1 + c) = 0 W.E-1: When the axes are translated to the point (-2,3) then the transformed equation of the curve 2 2 2x 4xy 5y 4x 22y 7 0       is Sol : a h b g f c         2, 2, 5, 2, 11, 7   2 2 hf bg gh af , 2, 3 ab h ab h             = x y 1 1 ,  Transformed equation is aX2 + 2hXY + bY2 + (gx1 + fy1 + c) = 0      2 2 2X 4XY 5Y 2 2 11 3 7 0             2 2 2X 4XY 5Y 22 0 v) To remove the first degree terms from the equation ax2 + by2 + 2gx + 2fy + c = 0, the origin is to be shifted to the point         b f a g , . In this case, the transformed equation is aX2 + bY2 + 0 2 2             c b f a g W.E-2: When the axes are translated to the point (-1,1) then the transformed equation of the curve 2 2 x 2y 2x 4y 2 0      is Sol : a b g f      1, 2, 1, 2 , c=2 new origin =   g f , 1, 1 a b           Transformed equation is aX2 + bY2 + 0 2 2             c b f a g    2 2 X 2 Y 1 2 2 0          2 2 X 2Y 1  Change of axes or transformation of axes is of three types : i) Translation of axes ii) Rotation of axes iii)General Transformation Translation of axes:  i) Shifting the origin to some other point without changing the direction of axes. ii) When the origin is translated to (h,k), the equa- tions of transformation are x = X+h, y =Y+k where (x, y) are the original coordinates and (X, Y) are the new coordinates of the point. Rotation of axes:  i) Rotating the system of coordinate axes through an angle ‘ ’ without changing the position of the origin. ii) When the axes are rotated through an angle ‘ ’ in anticlockwise direction. The equations of transformation are given by Set-1 x = X cos - Y sin y = Xsin + Ycos Set-2 X = xcos + ysin Y = -x sin + ycos  Transformation is used in reducing the general equation of any curve to the desired form. For example i) To eliminate first degree terms, we apply translation. ii) To eliminate the term containing ‘xy’, we apply rotation. iii) The point to which the origin has to be shifted to eliminate first degree terms (x, y terms) in S = ax2 + 2hxy + by2 + 2gx + 2fy +c=0 is obtained by solving 0,  0      y S x S TRANSFORMATION OF AXES SYNOPSIS
TRANSFORMATION OF AXES 4 4 MAINS ADVANCED MATHS BY ROHIT SIR MOB. 8218498992 [email protected] vi) To remove the first degree terms from 2hxy + 2gx + 2fy + c = 0, the origin is to be shifted to the point         h g h f , . In this case, the transformed equation is 2 2h XY 2gf ch 0    W.E-3 : When the origin is shifted to the point (5, -2) then the transformed equation of the curve xy 2x 5y 11 0     is Sol : 1 5 , 1, , 11 2 2 h g f c        f g , 5, 2 h h           Transformed equation is 2 2h XY 2fg ch 0    XY = - 21 vii) The point to which the origin has to be shifted to eliminate x and y terms in the equation       2 2 a x b y c is ,         W.E-4 : The point to which the origin has to be shifted to eliminate x and y terms in the equation 2 2 2x 3y 12x 12y 21 0      is Sol : First method : 2 2 2 3 12 12 21 0 x y x y      a b g f       2, 3, 6, 6  New origin = , 3, 2   g f a b          Second method : 2 2 2x 3y 12x 12y 21 0      2(x2 - 6x) + 3 (y2 - 4y) + 21 = 0 2(x2 - 6x+9) + 3(y2 -4y+4) - 18-12 + 21 = 0      2 2 2 x 3 3 y 2 9     Comparing with     2 2 a x b y c       , we get       3, 2 New Origin =     ,  = (3, 2) viii) a) To remove xy term of 2 2 ax hxy by gx fy c       2 2 2 0 the angle of rotation of axes is 1 2 1 T an 2 h a b           , if a b  2n 1 ,n z  4     if a = b b) If ' '  is angle of rotation to eleminate XY term in 2 2 ax hxy by gx fy c       2 2 2 0 , then 2 n   , n Z is also an angle of rotation to eliminate XY term ix) The angle of rotation of axes so that the equation ax + by + c = 0 is reduced as a) X = constant is 1 Tan b a        b) Y = constant is 1 Tan a b         W.E.-5 : The angle of rotation of the axes so that the equation 3 5 0 x y    may be reduced to the form y = constant is Sol : 3 5 0 x y    a b    3, 1  1 1 Tan Tan 3 3 a b               x) The equation S  ax2 +2hxy+by2 +2gx+2fy+c=0 has trans- formed to AX2 +2HXY+BY2 +2GX + 2FY + C = 0, when the origin is shifted to l m,  then A = a ; B = b ; H = h ; ( , ) 2 l m x S G          ( , ) 2 l m y S F            C = S l m,   The condition that the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 to take the form aX2 + 2hXY + bY2 = 0 when the axes are translated is abc + 2fgh - af2 - bg2 -ch2 = 0 General Transformation:  i) Applying both translation and rotation. ii) The equations of general transformation are given by
4 5 MAINS ADVANCED TRANSFORMATION OF AXES MATHS BY ROHIT SIR MOB. 8218498992 [email protected] Note : 1) If the rotation is in clockwise direction then replace  by - . 2) On translation or rotation the position of the point, length of line segment, area, perimeter, angles are not changed. But the coordinates and equations will change. 3) If the axes are rotated through an angle  the following are the some invariants in original and transformed equations of second degree general equation. 1) a + b 2) 2 h ab  3) c 4)   2 2 a b 4h   5) 2 2 2fgh af bg   6) 2 2 2 abc 2fgh af bg ch      If the Origin is shifted at O h k  ,  and the axis are rotated about the new origin O by an angle  in anticlock wise sense, then x h X Y    cos sin  ,y k X Y    sin cos   Proof : If O is shifted to O h k  ,  then new co-ordinates of p(x,y) become   1 1 x y, 1 1 x h x y k y     , Axis are rotated through an angle  , in the anticlockwise sense than the new co-ordinates of P  1 1 x y, become ( X,Y ) 0 x-axis X-axis ' y axis ' x axis P x ,y   x y axis Y axis   O ' 1 1      x X Y y X Y cos sin , sin cos            x h X Y y k X Y cos sin , sin cos      If by change of axes without change of origin the expression 2 2 ax hxy by   2 becomes 2 2 1 1 1 a X h XY bY   2 then i) 1 1 a b a b    ii) 2 2 1 1 1 ab h a b h    iii)    2 2 2 2 1 1 1 a b h a b h      4 4 Proof : Let the axes be rotated through an angle  in anticlockwise sense  x X Y   cos sin  , y X Y   sin cos          2 2 2 2 cos sin 2 cos sin sin cos sin cos ax hxy by a X Y h X X X Y b X Y                  2 2 But ax hxy by   2 2 2 1 1 1 transformed into a X h XY b Y   2 2 2 1     a a h b cos sin 2 sin    1 2 2 cos 2 sin 2 sin 2 h h a b       2 2 1 b a h b    sin sin 2 cos    by using these conditions, i) 1 1 a b a b    ii) 2 2 1 1 1 ab h a b h    iii)     2 2 2 2 1 1 1 a b h a b h      4 4 Translation of axes : 1. If (2, 3) are the coordinates of a point P in the new system when the origin is shifted to (-3, 7) then the original coordinates of P are 1) (-1, 10) 2) (5, -4) 3) (-5, 4) 4) (-1, 5) 2. The coordinates of the point (4,5) in the new system, when its origin is shifted to (3,7) are 1) (1, 2) 2) (-1, 2) 3) (-1, -2) 4) (1, -2) 3. When the origin is shifted to a point P, the point (2, 0) is transformed to (0, 4) then the coordinates of P are LEVEL - I (C.W)
TRANSFORMATION OF AXES 4 6 MAINS ADVANCED MATHS BY ROHIT SIR MOB. 8218498992 [email protected] ing origin then the coordinates of  2, 4 in old system are 1) 1 2 2,1 2 2    2) 1 2 2,1 2 2    3) 2 2, 2  4) 2, 2  12. If the axes are rotated through an angle 30o , the coordinates of 2 3, 3  in the new system are 1)        2 5 , 2 3 2)          2 5 , 2 3 3)          2 5 3 , 2 3 4)          2 5 3 3 2, 13. The transformed equation of x2 + 2 3 xy - y2 - 8 = 0, when the axes are rotated through an angle 6  is 1) X2 - Y2 = 0 2) X2 - Y2 = 4 3) X2 - Y2 = 2 4) X2 + Y2 = 4 14. If the axes are rotated through an angle 180o then the equation 2x - 3y + 4=0 becomes 1) 2X - 3Y - 4 = 0 2) 2X + 3Y - 4 = 0 3) 3X - 2Y + 4 = 0 4) 3X + 2Y + 4 = 0 15. If the transformed equation of a curve is 17X2 -16XY + 17Y2 = 225 when the axes are rotated through an angle 45o , then the origi- nal equation of the curve is 1) 25x2 + 9y2 = 225 2) 9x2 + 25y2 = 225 3) 25x2 - 9y2 = 225 4) 9x2 - 25y2 = 225 16. If the equation 4x2 + 2 3 xy + 2y2 - 1 = 0 becomes 5X2 + Y2 = 1, when the axes are rotated through an angle  , then  is 1) 15o 2) 30o 3) 45 o 4) 60 o 17. The angle of rotation of axes in order to eliminate xy term in the equation xy = c2 is 1) 12  2) 6  3) 3  4) 4  18. The transformed equation of 2 2 2 x y r   , when the axes are rotated through an 1) (2, -4) 2) (-2, 4) 3) (-2, -4) 4) (2, 4) 4. If the axes are translated to the point (-2,-3) then the equation x2 +3y2 +4x+18y+30=0 transforms to 1) X2 + Y2 = 4 2) X2 + 3Y2 =1 3) X2 - Y2 = 4 4) X2 - 3Y2 = 1 5. When the axes are translated to the point (5, -2) then the transformed form of the equa- tion xy + 2x - 5y - 11 = 0 is 1) X 1 Y  2) Y 1 X  3) XY = 1 4) XY2 = 2 6. If the transformed equation of a curve when the origin is translated to (1, 1) is 2 2 X Y X Y      2 2 0 then the original equation of the curve is 1) 2 2 x y   2 1 2) 2 2 x y y     3 3 0 3) 2 2 x y y     3 3 0 4) 2 2 x y y     3 3 0 7. In order to make the first degree terms missing in the equation 2x2 + 7y2 + 8x - 14y + 15 = 0, the origin should be shifted to the point 1) (1, -2) 2) (-2, -1) 3) (2, 1) 4) (-2, 1) 8. The point to which the origin should be shifted in order to remove the x and y terms in the equation 14x2 - 4xy + 11y2 - 36x + 48y + 41 = 0 is 1) (1, -2) 2) (-2, 1) 3) (-1, 2) 4) (2, -1) 9. If the distance between the two given points is 2 units and the points are transferred by shifting the origin to (2, 2), then the distance between the points in their new position is 1) 2 2) 5 3) 6 4) 7 10. When (0, 0) shifted to (3, -3) the coordinates of P(5, 5), Q(-2, 4) and R(7, -7) in the new system are A, B, C then area of triangle ABC in sq units is 1) 43 2) 23 3) 45 4) 50 Rotation of axes : 11. When axes are rotated through an angle of 450 in positive direction without chang-