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JEE • MATHEMATICS • TRANSFORMATION OF AXES

Transformation of axes practice · Mixed difficulty

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Showing 31-38 of 38 questions

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31 Easy
31 of 38
Analyzing +4 | -1
Let 0≤θ≤π20 \le \theta \le \frac{\pi }{2} and x=Xcos⁡θ+Ysin⁡θ,y=Xsin⁡θ−Ycos⁡θx = X\cos \theta + Y\sin \theta ,y = X\sin \theta - Y\cos \theta such that x2+4xy+y2=aX2+bY2,{x^2} + 4xy + {y^2} = a{X^2} + b{Y^2},, where a,ba,b are constants, then
View answer and explanation
Correct option: C
a=3,  b=−1,  θ=π4a = 3,\;b = - 1,\;\theta = \frac{\pi }{4}
Option C (a=3,  b=−1,  θ=π4a = 3,\;b = - 1,\;\theta = \frac{\pi }{4}) is correct because substituting these values into the given equation satisfies the condition.
32 Medium
32 of 38
Analyzing +4 | -1
If x=Xcos⁡θ−Ysin⁡θ,y=Xsin⁡θ+Ycos⁡θx = X\cos \theta - Y\sin \theta ,y = X\sin \theta + Y\cos \theta and x2+4xy+y2=AX2+BY2,0≤θ≤π2,{x^2} + 4xy + {y^2} = A{X^2} + B{Y^2},0 \le \theta \le \frac{\pi }{2}, then
View answer and explanation
Correct option: B
θ=π4\theta = \frac{\pi }{4}
Option B (θ=π4\theta = \frac{\pi }{4}) is correct because substituting the given transformations into the equation and comparing coefficients leads to the condition that θ=π4\theta = \frac{\pi}{4}.
33 Easy
33 of 38
Application +4 | -1
To which point should the origin be shifted so that the coordinates of the point (6,−1)(6,-1) become (8,−4)(8,-4) ?
View answer and explanation
Correct option: D
(2,3)(2,3)
Option A ((2,3)(2,3)) is incorrect. The correct answer is (−2,3)(-2,3) because it represents the shift in origin needed to transform the coordinates as specified. The provided solution has a sign error.
34 Medium
34 of 38
Application +4 | -1
If the equation 2x2+y2−4x−4y=02{x^2} + {y^2} - 4x - 4y = 0 is transformed to the equation 2X2+Y2−8X−8Y+18=02{X^2} + {Y^2} - 8X - 8Y + 18 = 0 by shifting the origin at a point PP without rotating the coordinates axes, then the coordinates of PP are
View answer and explanation
Correct option: D
(−1,−2)\left( { - 1, - 2} \right)
Option D ((-1,-2)) is correct because it represents the shift in origin that transforms the first equation into the second.
35 Easy
35 of 38
Application +4 | -1
In order to remove first degree terms from the equation 2x2+7y2+8x−14y+4=0,2{x^2} + 7{y^2} + 8x - 14y + 4 = 0, the origin is shifted at the point
View answer and explanation
Correct option: A
(−2,  1)\left( { - 2,\;1} \right)
Option A ((-2, 1)) is correct because shifting the origin to this point removes the first-degree terms from the given equation.
36 Easy
36 of 38
Application +4 | -1
The pair of lines 3x2−4xy+3y2=0\sqrt 3 {x^2} - 4xy + \sqrt 3 {y^2} = 0 are rotated about the origin by π/6\pi /6 in the anti-clockwise sense. The equation of the pair in the new position is
View answer and explanation
Correct option: A
3y2−xy=0\sqrt 3 {y^2} - xy = 0
Option A (3y2−xy=0\sqrt 3 {y^2} - xy = 0) is correct because after rotating the pair of lines by π/6\pi/6 in the anti-clockwise sense, the equation becomes 3y2−xy=0\sqrt 3 {y^2} - xy = 0.
37 Easy
37 of 38
Application +4 | -1
The new coordinates of a point (4,5)(4,5), when the origin is shifted to the point (1,−2)(1,-2) are
View answer and explanation
Correct option: C
(3,7)(3,7)
Option C ((3,7)(3,7)) is correct because when the origin is shifted to (1,−2)(1,-2), the new coordinates are found by subtracting the new origin's coordinates from the original point's coordinates: (4−1,5−(−2))=(3,7)(4-1, 5-(-2)) = (3,7).
38 Easy
38 of 38
Application +4 | -1
At what point the origin be shifted, if the coordinates of a point (4,5)(4,5) become (−3,9)(-3,9) ?
View answer and explanation
Correct option: B
(7,−4)(7,-4)
Option B (7, -4) is correct because shifting the origin involves subtracting the new coordinates from the old coordinates to find the shift.
Revision notes and key takeaways

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