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Madeleine Richard
6 years ago
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1 COORDINATE GEOMETRY CHANGE OF AXES EXERCISE 1. The point to which the origin should be shifted in order to eliminate x and y terms in the equation 4x 9y 8x 36y 4 0 is (1, ) ( 4, ( 1,) 4) (1, ). In order to eliminate the first degree terms from the equation x 4xy 5y 4x y 7 0, the point to which origin is to be shifted is (1, ) (, (, 4) (1, 3. The point to which the origin should be shifted in order to eliminate x and y terms in the equation 14x 4xy 11y 36x 48y 41 0 is (1, ) ( 4, ( 1,) 4) (1, ) 4. The point to which the axes are to translated to eliminate y term and constant term in the equation y +8x+4y = 0 is (3, ) ) (3, / (3/4, ) 4) (/3, 4) 5. If the axes are tranlated to the circumcentre of the triangle formed by (9,, ( 1,7), ( 1, then the centroid of the triangle in the new system is (5,5/ ) (4, ( 5/3, / 4) (0,0) 6. The transformed equation of x y x 4y 0 when the axes are translated to the point ( 1, is. X Y 1 ) X 3Y 1 X Y 3 0 4) 4X 9Y If the first degree terms of x 4xy y x y 6 0 are eliminated by translation of axes then the transformed equation is. X 4XY Y 8 ) X 4XY Y 6 X 4XY Y 4 4) X 4XY Y 8. If the transformed equation of a curve is 3X XY Y 7X Y 7 0 when the axes are translated to the point (1,), then the original equation of the curve is 3x xy y 15x 4y 13 0 ) 3x xy y 15x 4y x xy y 15x 4y ) 3x xy y 15x 4y The origin is shifted to (1,). The equation y 8x 4y 1 0 changes to y 4ax then a = 1 ) 4) 1

2 10. By translating the axes the equation xy x+y = 6 has changed to xy = c, then c = 4 ) 5 6 4) If the axes are rotated through an angle 45, the coordinates of, 3 in the new system are 3 3, 5 ) ( 1, 5) 5 3, 7 4) If the coordinates of a point P are transformed to 4, 6 3 when the axes are rotated through an angle 30, then P = 3 3, 5 ) ( 1, 5) 5 3, 7 4) If the axes are rotated through an angle 45 in the positive direction without changing the origin, then the coordinates of the point, 4 in the old system are 1,1 ) 1,1, 4), 14. The angle of rotation of axes in order to eliminate xy term in the equation x 3xy y a is / 6 ) / 4 / 3 4) / 15. The angle of rotation of axes to remove xy term in the equation 9x 3xy 3y 0 is / 6 ) / 4 / 3 4) 5 /1 16. The angle of rotation of axes to remove xy term in the equation x 4xy y x y 6 0 is /1 ) / 6 / 3 4) / The transformed equation of x 6xy 8y 10 when the axes are rotated through an angle / 4 is 15x 14xy 3y 0 ) 15x 14xy 3y 0 15x 14xy 3y 0 4) 15x 14xy 3y If the axes are rotated through an angle 30 about the origin then the transformed equation of x 3xy y a is.

3 X Y a ) X Y a X Y a 4) X Y a 19. The transformed equation of ax hxy by gx fy c 0 when the axes are rotated through an angle 90 is bx hxy ay fx gy c 0 ) bx hxy ay fx gy c 0 bx hxy ay fx gy c 0 4) bx hxy ay fx gy c 0 0. The transformed equation of x y 4x 6y 1 0 when the axes are rotated through an angle 180 is X Y 4X 6Y 1 0 ) X Y 4X 6Y 1 0 X Y 4X 6Y 1 0 4) X Y 4X 6Y If the transformed equation of a curve is 17X 16XY 17Y 5 when the axes are rotated through an angle 45, then the original equation of the curve is 5x 9y 5 ) 9x 5y 5 5x 9y 5 4) 9x 5y 5. If the transformed equation of a curve is X XY tan Y a when the axes are rotated through an angle, then the original equation of the curve is x y a cos ) x y a cos x a y cos 4) x a y cos 3. The angle of rotation of the axes so that the equation 3x y 5 0 may be reduced to the form Y = constant is / 6 ) / 4 / 3 4) / 4. A line L has intercepts a and b on the coordinate axes. Keeping the origin fixed, the axes are rotated through a fixed angle. Then the same line has intercepts p and q on the new axes. Then a p b q ) a b p q a b p q 4) a p b q 5. The line joining two points A(,0), B(3, is rotated about A anticlock wise direction through an angle 15. If B goes to C then C =

4 1 3, ) 1 3, 1, 3 4) 1, 3 6. The point (4, undergoes the following three transformations successively i) reflection about the line y = x ii) translation through a distance unit along the positive direction of x axis. The final position of the point is (3,4) ) (4, ( 1,4) 4) none 7. The point (4, undergoes the following three transformations successively (i) Reflection about the line y = x (ii) Translation through a distance unit along the positive direction of x-axis. (iii) Rotation through an angle / 4about the origin in the clockwise direction. The final position of the point is given by the coordinates 1/,7 / ),7 1,7 / 4),7 8. The point P(1, is translated parallel to x = y in the first quadrant through a unit distance. The coordinates of the new position of P are 1 1, , ) 1, ) 1, 5 5

5 CHANGES OF AXES SOLUTIONS 1. Ans.4 g f 4 18 Here a = 4, b = 9, g = -4, f = 18. Required point is., 1, a b 4 9. Ans.3 hf bg gh af Required point =, ab h ab h Ans Ans =,3 Here a = 14, h = -, b = 11, g = -18, f = 4 Required point is hf by gh af ,, ab h ab h =, 1 Given equation is (3/4,-) 5. Ans.3 y 8x 4y 0 y 8 x 3/ 4 0 Point of translation = Given points from a right angled triangle right angled at (-1, Circumcentre = Midpoint of (9,, (-1,7) = (4,5) Centroid of the triangle = (7/3, 13/ Centroid in the new system = 4, 5, Ans.1 x = X-1, y = Y+1 The transformed equation is (X- +(Y+ +(X--4(Y++=0 X +Y =0

6 7. Ans Point of translation =, 1, X 4XY Y X 4XY Y 4 Transformed equation is 8. Ans. X x 1, Y y The original equation is 3 x 1 x 1 x 1 y y 7 x 1 y Ans. x=x+1, y=y+ The transformed equation of y 8x 4y 1 0 is 3x xy y 15x 4y 13 0 Y 8 X 1 4 Y 1 0 Y 8X 0 Y 8X a 10. Ans.1 x=x+h, y=y+k 11. Ans. The transformed equation of xy x 6 is X h Y k X h Y k 6 0 XY k 1 X h Y hk h k 6 0 comparing this equation with xy = c we get k 1 = 0, h + = 0 & c = -(hk-h+k-6) k = 1, h = - & c = -[(-)(++-6]=4 x, y, X 3 1, Y 3 5 X, Y 1, 5 1. Ans.3

7 X, Y 4, 6 3, y Xsin Y cos y Xsin Y cos Ans Ans.1 P 5 3, 7 x cos 45 4sin 45 1/ 4 1/ y sin 45 4cos 45 1/ 4 1/ 1 Required point = 1,1 Comparing the given equation with ax hxy by c we get a = 1, h 3, b Ans.4 Angle of rotation is 1 h a b Tan Tan Tan Here a = 9, b = 3, h = 3 1 h Angle of rotation, Tan Tan Tan a b n 5 when n = Ans.4

8 Coefficient of x 1 coefficient of y. 17. Ans.3 Angle of rotation is / 4 (X,Y) be the new coordinates of (x,y) when the axes are rotated through an angle / Ans. 19. Ans.1 X Y X Y Then x, y X Y X Y X Y X Y The transformed equation is X Y XY 6 X Y 8 X Y XY 0 15X 14XY 3Y 0 x X cos30 Ysin 30 3X Y X 3Y y Xsin 30 Y cos30 Given equation is x 3xy y a 3X Y 3X Y X 3Y X 3Y 3 a 3X Y 3XY 3 3X 3XY XY 3Y X 3Y 3XY 8a X Y 4 3XY 6X 6 3XY 3XY 6y = 8a X Y 4 3XY 6X 6 3XY 3XY 6Y 8a 8X 8Y 8a X Y a x X cos90 Y sin 90 Y, y X sin 90 Y cos90 X The transformed equation is a Y h Y X bx g Y f X C 0 bx hxy ay fx gy c 0 0. x X cos180 Y sin180 X, y Xsin180 Y cos180 Y

9 The transformed equation is 1. Ans. 1. Ans. X Y 4X 6Y 1 0 y x Y x sin 45 ycos 45 x y X x cos 45 ysin 45, The original equation of the curve is X Y 4 X 6 Y 1 0 x y x y y x y x x y y x y x 5 5 5x 9y 5 X x cos y sin, Y x sin y cos The original equation of the curve is x cos ysin x cos ysin x sin ycos tan x sin ycos a x cos y sin xy cos sin xy sin tan y cos sin tan x x cos sin tan xy cos tan sin y cos xy sin cos a x cos sin tan sin y sin cos tan sin tan sin sin x y cos a cos sin sin tan cos xy 3. Ans.3 x y a cos a

10 1 a 1 3 Angle of rotation = Tan Tan b Ans.3 x y 1 1 a b Equation of L with a,b as intercepts is 5. Ans.1 Transformed equation to L with p, q as intercepts is The distance from origin to ( and () is the same a b 1 1 a b p q p q X Y 1 p q By changing the origin to A(,0) the coordinates of B become (1,. 6. Ans Ans.3 Now 15 1 X cos15 sin15, Y sin15 cos Changing the origin to the original place, then the coordinates of C are, Reflection of (4, with reference to y = x is (1,4) The point (1,4) is translated through distance along a horizontal line in the direction of x axis. The new coordinates of (1,4) are (3,4). Reflection of (4, about the line y = x is (1,4). Translation through a distance along positive direction of x-axis is (3,4). Rotation through an angle / 4 in the clockwise direction is 3 3cos 4sin 3sin 4 cos = 1 7, 8. Ans. If is the inclination of the given line then 1 tan cos,sin 5 5

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3 Drill Exercise - 1 1. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ). 2. Prove that the points (2a, 4a) (2a, 6a) and (2a + 3 a, 5a) are the vertices of an equilateral