# Rotation of Axes

## Trending Questions

**Q.**

Y = A sin ( ωt - kx ). Write the dimensions of w and k if x is distance and t is time.

**Q.**A line has intercepts a and b on the coordinate axes. When the axes are rotated through an angle α in anticlockwise direction, keeping the origin fixed, the line makes equal intercepts on the coordinate axes. Then the value of cotα is

- a+ba−b
- a−ba+b
- a2−b2
- a2+b2

**Q.**If the axes are shifted to (−2, −3) and then rotated through π4 in anticlockwise direction, then transformed equation of x2−y2+2x+4y=0 is

- 8x+12y−2√2xy−21√2=0
- 8x−12y−2√2xy−21√2=0
- 8x+12y+2√2xy−21√2=0
- 8x+12y+2√2xy+21√2=0

**Q.**The transformed equation of ax2+2hxy+by2+2gx+2fy+c=0 when the axes are rotated through an angle of 90∘ is

- bX2−2hXY+aY2+2fX−2gY+c=0
- bX2+2hXY+aY2+2fX+2gY+c=0
- bX2−2hXY+aY2−2fX+2gY+c=0
- bX2−2hXY+aY2−2fX−2gY+c=0

**Q.**The transformed equation of 3x2+3y2+2xy=2 when the coordinate axes are rotated through an angle 45∘ is

- X2+2Y2=1
- 2X2+Y2=1
- 2X2−Y2=1
- X2−2Y2=1

**Q.**If the coordinate of a point P are transformed to (4, −6√3) when the axis are rotated through an angle 30 ∘ in the anti clockwise direction, then original coordinates of P are

- (√3, 4)
- (5√3, 7)
- (5√3, −7)
- (4√3, −7)

**Q.**If the axes are rotated through an angle of 90∘ in any direction then the transformed equation of x2=4ay can be

- Y2=4aX
- Y2=−4aX
- X2=4aY
- X2=−4aY

**Q.**The transformed equation of x2+6xy+8y2=10 when the axes are rotated through an angle π4 (in the anti clockwise direction) is aX2+2hXY+bY2=20 then which of the following is/are correct

- a+b=18
- h=14
- h=7
- a−b=12

**Q.**The transformed equation of 9x2+2√3xy+7y2=10 when the axes are rotated through an angle of π6 (in the anti clockwise direction) is

- 5X2+3Y2=5
- 5X2−2Y2=5
- 4X2+3Y2=6
- 3X2−Y2+2√2Y−6=0

**Q.**The equation of a curve is 3x2+2xy+3y2=10

. Its equation if the axes are rotated through an angle 45° will be

- 2x2+y2=5
- 2x2+y2=10
- x2+2y2=5
- x2+2y2=10

**Q.**The transformed equation of ax2+bxy+cy2=2 is 2X2+Y2=1, when the axes are rotated through an angle of 45∘ in anticlockwise direction. Then which of the following is (are) true?

- a2+b2+c2=20
- Roots of the equation am2+bm+c=0 are real
- Arithmetic mean of a and c is b+1
- Range of the function f(y)=ay−bby−c is R−{32}

**Q.**

The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p and q are the intercepts of the line L on the new axes, then 1a2−1p2+1b2−1q2 is equal to

0

1

none of these

-1

**Q.**

In the Argand plane, the vector z=4-3i is turned in the clockwise sense through 180o and stretched three times. The complex number represented by the new vector is

12-9i

-12-9i

-12+9i

12+9i

**Q.**

A rod of length 12 cm
moves with its ends always touching the coordinate axes. Determine
the equation of the locus of a point P on the rod, which is 3 cm from
the end in contact with the *x*-axis.

**Q.**Let S be the circle in xy-plane which touches the x-axis at point A, the y-axis at point B and the unit circle x2+y2=1 at point C externally. If O denotes the origin, then the angle OCA equals

- 3π5
- 5π8
- π2
- 3π4

**Q.**The locus of the mid-point of the intercept of the variable line xcosα+ysinα=p (p is constant) between the coordinates axes is

- 1x2+1y2=1p2
- None of these
- 1x2+1y2=2p2
- 1x2+1y2=4p2

**Q.**The transformed equation of 9x2+2√3xy+7y2=10 when the axes are rotated through an angle of π6 (in the anti clockwise direction) is

- 3X2−Y2+2√2Y−6=0
- 5X2+3Y2=5
- 5X2−2Y2=5
- 4X2+3Y2=6

**Q.**If the transformed equation of xy=c2 when the axis are rotated through an angle of π4 (in the anti clockwise direction), is pX2+qY2=rc2, then p−q+r is equal to

- 2
- 0
- 8
- 4

**Q.**If the transformed equation of xy=c2 when the axis are rotated through an angle of π4 (in the anti clockwise direction), is pX2+qY2=rc2, then

- p=1
- p=−1
- q=1
- r=2

**Q.**If the axes are shifted to (−2, −3) and then rotated through π4 in anticlockwise direction, then transformed equation of x2−y2+2x+4y=0 is

- 8x+12y−2√2xy−21√2=0
- 8x−12y−2√2xy−21√2=0
- 8x+12y+2√2xy−21√2=0
- 8x+12y+2√2xy+21√2=0

**Q.**If the transformed equation of xy=c2 when the axis are rotated through an angle of π4 (in the anti clockwise direction), is pX2+qY2=rc2, then

- p=1
- p=−1
- q=1
- r=2

**Q.**The transformed equation of ax2+2hxy+by2+2gx+2fy+c=0 when the axes are rotated through an angle of 90∘ is

- bX2−2hXY+aY2+2fX−2gY+c=0
- bX2+2hXY+aY2+2fX+2gY+c=0
- bX2−2hXY+aY2−2fX+2gY+c=0
- bX2−2hXY+aY2−2fX−2gY+c=0

**Q.**The axes being inclined at an angle ω, find the centre and radius of the circle x2+2xycosω−2gx−2fy=0.

**Q.**

Final coordinates of a point as a result of rotating a point about origin through an angle of θ is equivalent to rotating the co-ordinate axes through an angle of −θ.

True

False

**Q.**

Final coordinates of a point as a result of rotating a point about origin through an angle of θ is equivalent to rotating the co-ordinate axes through an angle of −θ.

True

False

**Q.**

The line L has intercepts a and b on the coordinate axes. When keeping the origin fixed, the coordinate axes are rotated through a fixed angle, then the same line has intercepts p and q on the rotated axes. Then

(I.I.T. 1990)

a2+b2=p2+q2

1a2+1b2=1p2+1q2

a2+b2=p2+q2

1a2+1p2=1b2+1q2

**Q.**The locus of the mid point of the intercept of the line xcosα+ysinα=p between the coordinate axes is

- x−2+y−2=4p−2
- x2+y2=4p−2
- x−2+y−2=p−2
- x2+y2=p−2

**Q.**lf the axes are rotated through an angle 60o, then the transformed equation of x2+y2=25 is

- X2+Y2=1
- X2+Y2=9
- X2+Y2=16
- X2+Y2=25

**Q.**

The origin of the co-ordinate axes is shifted to (-1, 3) and the axes is rotated through an angle of 90∘ in anti-clockwise direction. If (a, b) is the new coordinates of (2, 3) in the new coordinate system, then find the value of 2a2+3b2

**Q.**The transformed equation of 3x2+3y2+2xy=2 when the coordinate axes are rotated through an angle 45∘ is

- X2+2Y2=1
- 2X2+Y2=1
- X2−2Y2=1
- 2X2−Y2=1