Shifting of Axes

Q. The transformed equation of $x^2+2y^2+2x-4y+2=0$ when origin is shifted to the point $(-1,1)$ with out rotation of axes, is

1. $X^2-2Y^2=1$
2. $2X^2+Y^2=1$
3. $X^2+2Y^2=1$
4. $X^2-Y^2=0$

Q. If the origin is shifted to (7, -4), then the coordinates of the point P(4, 5) will become .

1. (-3, 9)
2. (-3, -9)
3. (3, 9)
4. (3, -9)

Q.

Write the distance between the vertex and focus of the parabola $y^2+6y+2x+5=0.$

Q.

$\text{Find what the following equations become when the origin is shifted to the point(1, 1)}?\left(i\right)x^2+xy-3y^2-y+2=0\left(ii\right)xy-y^2-x+y=0\left(iii\right)xy-x-y+1=0\left(iv\right)x^2-y^2-2x+2y=0$

Q.

Axis of a parabola is $y=x$ and vertex and focus are at a distance $\sqrt{2}$ and $2\sqrt{2}$, respectively from the origin. Then, equation of the parabola is

1. ${(x–y)}^2=8(x+y–2)$

2. ${(x+y)}^2=2(x+y–2)$

3. ${(x–y)}^2=4(x+y–2)$

4. ${(x+y)}^2=2(x-y+2)$

Q.

A rod of length I sides with its ends on two perpendicular lines. Then, the locus of its midpoint is

1. 

2. 

3. 

4. 

Q. A cube of side 3 units has one vertex at point (1, 1, 1) and the three edges from this vertex are respectively parallel to positive x - axis and negative y and z - axes. Find the coordinates of other vertices of the cube.

Q. The transformed coordinates of the point $(4,3)$ when the axes are translated to point $(3,1)$ and then rotated through ${30}^{\circ }$ in anticlockwise direction is

1. $(\frac{2\sqrt{3}+1}{2},\frac{\sqrt{3}-2}{2})$
2. $(\frac{\sqrt{3}+1}{2},\frac{2\sqrt{3}+1}{2})$
3. $(\frac{\sqrt{3}+2}{2},\frac{2\sqrt{3}-1}{2})$
4. $(\frac{\sqrt{3}-2}{2},\frac{\sqrt{3}+1}{2})$

Q.

Find the area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant.

Q.

Find the area of the region bounded by x² = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Q.

verify that the area of the triangle with vertices (2, 3) , (5, 7) and (-3-1) aremains invariant under the translation of axes when the origin is shifted to the point(-1, 3).

Q. By shifting origin to a suitable point with out rotation of axes, the equation $xy-x+2y=6$ has transformed to $XY=B$, then the value of $B$ is

Q. x - axis is a tangent and y - axis is normal to a parabola whose focus is (2, 3)
The equation of tangent at vertex of parabola is

1. 2x - 3y+9 = 0
2. 2x - 3y - 4 = 0
3. 3x + 2y - 6 = 0
4. 3x + 2y + 1 = 0

Q.

Verify that area of the triangle with vertices (4, 6) (7, 10) and(1, -2) remains invariant under the translation of axes when the origin is shifted to the point (-2, 1).

Q. If $(7,5)$ are the new coordinates of $P$ when origin is shifted to $(1,1)$, then the original coordinates of $P$ are

1. $(8,6)$
2. $(6,4)$
3. $(-6,-4)$
4. $(4,3)$

Q.

In the new coordinate system origin is shifted to (h, k) and the axes are rotated through angle of ${90}^{\circ }$ in the anti-clockwise direction. The new co-ordinates of (x, y) is obtained by the following method.

1) (x + iy) becomes (x + iy) $e^{\frac{-i\pi }{2}}$

Let it be (x+iy) or (x, y)

2)(x, y) becomes (x - h, y - k)

1. True

2. False

Q.

$\text{What does the equation}(a-b)(x^2+y^2)-2abx=0becomeiftheoriginisshiftedtothepoint(\frac{ab}{a-b},0)withoutrotation?$

Q.

$\text{Find the point to which the origin should be dhifted after a translation of axes so that the following equations will have no first degree terms:}?\left(i\right)y^2+x^2-4x-8y+3=0\left(ii\right)x^2+y^2-5x+2y-5=0\left(iii\right)x^2-12x+4=0$

Q.

The number of straight lines equally inclined to both the axes are

A) 1

B) 0

C) 2

D) Infinite

Q.

$\text{At what point the origin be shifted so that the equation}{x}^2+xy-3x-y+2=0\text{does not contain any first degree term and constant term ?}$

Q.

Write the equation of the direction of the parabola $x^2-4x-8y+12=0$.

Q. The point where the origin has to be shifted so that the equation $y^2+4y+8x-2=0$ will not contain $y$ and the constant term is

1. $(\frac{3}{4},-2)$
2. $(\frac{3}{5},-2)$
3. $(2,4)$
4. $(1,-4)$

Q.

Without changing the direction of coordinate axes, origin is transferred to (h, k), so that the linear (one degree)

terms in the equation $x^2+y^2-4x+6y-7=0$ are eliminated. Then the point (h, k) is

1. (3, 2)

2. (- 3, 2)

3. (2, - 3)

4. (1.7)

Q. Equation of the image of the pair of rays y = | x | by the line x = 1 is

1. | y | + 2 = x
2. none of these
3. y = | x - 2|
4. | y | = x + 2

Q. When the origin is shifted to $(1,2),$ the equation $y^2-8x-4y+12=0$ changes to $Y^2=4aX,$ then value of $a$ is

Q. The point where the origin has to be shifted so that the equation $y^2+4y+8x-2=0$ will not contain $y$ and the constant term is

1. $(\frac{3}{4},-2)$
2. $(\frac{3}{5},-2)$
3. $(2,4)$
4. $(1,-4)$

Q.

The new coordinates of a point (4, 5), when the origin is shifted to the point (1, -2) are

1. (3, 7)

2. None of these

3. (3, 5)

4. (5, 3)

Q.

If origin is shifted to the point $(2,3)$ without rotation of axes then the coordinates of the point $P$ which divides the join of $A(4,8)$ and $B(7,14)$ in the ratio $1:2$ or $2:1$ with respect to the new system of coordinates can be

1. $(2,5)$

2. $(3,7)$

3. $(4,9)$

4. $(5,11)$

Q.

$\text{Find what the following equations become when the origin is shifted to the point(1, 1)}?\left(i\right)x^2+xy-3y-y+2=0\left(ii\right)x^2-y^2-2x+2y=0\left(iii\right)xy-x-y+1=0\left(iv\right)xy-y^2-x+y=0$

Q. If the origin is shifted to (7, -4), then the coordinates of the point P(4, 5) will become .

1. (-3, 9)
2. (-3, -9)
3. (3, 9)
4. (3, -9)