Tiêu đề XI - maths - chapter 12 - TRANSFORMATION OF AXES (40-51).pdf ✅

TRANSFORMATION OF AXES JEE-MAIN-JR-MATHS VOL-III 40 NARAYANAGROUP  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 equations 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 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 2Y 1 2 2 0          2 2 X 2Y 1
JEE-MAIN-JR-MATHS VOL-III TRANSFORMATION OF AXES NARAYANAGROUP 41 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 transformed 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