Rotation of Axes

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 $\alpha$ in anticlockwise direction, keeping the origin fixed, the line makes equal intercepts on the coordinate axes. Then the value of $\cot \alpha$ is

1. $\frac{a+b}{a-b}$
2. $\frac{a-b}{a+b}$
3. $a^2-b^2$
4. $a^2+b^2$

Q. If the axes are shifted to $(-2,-3)$ and then rotated through $\frac{\pi }{4}$ in anticlockwise direction, then transformed equation of $x^2-y^2+2x+4y=0$ is

1. $8x+12y-2\sqrt{2}xy-21\sqrt{2}=0$
2. $8x-12y-2\sqrt{2}xy-21\sqrt{2}=0$
3. $8x+12y+2\sqrt{2}xy-21\sqrt{2}=0$
4. $8x+12y+2\sqrt{2}xy+21\sqrt{2}=0$

Q. The transformed equation of $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ when the axes are rotated through an angle of ${90}^{\circ }$ is

1. $b{X}^2-2hXY+a{Y}^2+2fX-2gY+c=0$
2. $b{X}^2+2hXY+a{Y}^2+2fX+2gY+c=0$
3. $b{X}^2-2hXY+a{Y}^2-2fX+2gY+c=0$
4. $b{X}^2-2hXY+a{Y}^2-2fX-2gY+c=0$

Q. The transformed equation of $3x^2+3y^2+2xy=2$ when the coordinate axes are rotated through an angle ${45}^{\circ }$ is

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

Q. If the coordinate of a point $P$ are transformed to $(4,-6\sqrt{3})$ when the axis are rotated through an angle $30{}^{\circ }$ in the anti clockwise direction, then original coordinates of $P$ are

1. $(\sqrt{3},4)$
2. $(5\sqrt{3},7)$
3. $(5\sqrt{3},-7)$
4. $(4\sqrt{3},-7)$

Q. If the axes are rotated through an angle of ${90}^{\circ }$ in any direction then the transformed equation of $x^2=4ay$ can be

1. $Y^2=4aX$
2. $Y^2=-4aX$
3. $X^2=4aY$
4. $X^2=-4aY$

Q. The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi }{4}$ (in the anti clockwise direction) is $a{X}^2+2hXY+b{Y}^2=20$ then which of the following is/are correct

1. $a+b=18$
2. $h=14$
3. $h=7$
4. $a-b=12$

Q. The transformed equation of $9x^2+2\sqrt{3}xy+7y^2=10$ when the axes are rotated through an angle of $\frac{\pi }{6}$ (in the anti clockwise direction) is

1. $5X^2+3Y^2=5$
2. $5X^2-2Y^2=5$
3. $4X^2+3Y^2=6$
4. $3X^2-Y^2+2\sqrt{2}Y-6=0$

Q. The equation of a curve is $3x^2+2\mathrm{xy}+3y^2=10$
. Its equation if the axes are rotated through an angle $45°$ will be .

1. $2x^2+y^2=5$
2. $2x^2+y^2=10$
3. $x^2+2y^2=5$
4. $x^2+2y^2=10$

Q. The transformed equation of $a{x}^2+bxy+c{y}^2=2$ is $2X^2+Y^2=1,$ when the axes are rotated through an angle of ${45}^{\circ }$ in anticlockwise direction. Then which of the following is (are) true?

1. $a^2+b^2+c^2=20$
2. Roots of the equation $a{m}^2+bm+c=0$ are real
3. Arithmetic mean of $a$ and $c$ is $b+1$
4. Range of the function $f\left(y\right)=\frac{ay-b}{by-c}$ is $R-\left\{\frac{3}{2}\right\}$

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 $\frac{1}{a^2}-\frac{1}{p^2}+\frac{1}{b^2}-\frac{1}{q^2}$ is equal to

1. 0

2. 1

3. none of these

4. -1

Q.

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

1. 12-9i

2. -12-9i

3. -12+9i

4. 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 $x^2+y^2=1$ at point C externally. If $O$ denotes the origin, then the angle $OCA$ equals

1. $\frac{3\pi }{5}$
2. $\frac{5\pi }{8}$
3. $\frac{\pi }{2}$
4. $\frac{3\pi }{4}$

Q. The locus of the mid-point of the intercept of the variable line $x\cos \alpha +y\sin \alpha =p$ ($p$ is constant) between the coordinates axes is

1. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{p^2}$
2. $Noneofthese$
3. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{2}{p^2}$
4. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$

Q. The transformed equation of $9x^2+2\sqrt{3}xy+7y^2=10$ when the axes are rotated through an angle of $\frac{\pi }{6}$ (in the anti clockwise direction) is

1. $3X^2-Y^2+2\sqrt{2}Y-6=0$
2. $5X^2+3Y^2=5$
3. $5X^2-2Y^2=5$
4. $4X^2+3Y^2=6$

Q. If the transformed equation of $xy=c^2$ when the axis are rotated through an angle of $\frac{\pi }{4}$ (in the anti clockwise direction), is $p{X}^2+q{Y}^2=r{c}^2$, then $p-q+r$ is equal to

1. $2$
2. $0$
3. $8$
4. $4$

Q. If the transformed equation of $xy=c^2$ when the axis are rotated through an angle of $\frac{\pi }{4}$ (in the anti clockwise direction), is $p{X}^2+q{Y}^2=r{c}^2$, then

1. $p=1$
2. $p=-1$
3. $q=1$
4. $r=2$

Q. If the axes are shifted to $(-2,-3)$ and then rotated through $\frac{\pi }{4}$ in anticlockwise direction, then transformed equation of $x^2-y^2+2x+4y=0$ is

1. $8x+12y-2\sqrt{2}xy-21\sqrt{2}=0$
2. $8x-12y-2\sqrt{2}xy-21\sqrt{2}=0$
3. $8x+12y+2\sqrt{2}xy-21\sqrt{2}=0$
4. $8x+12y+2\sqrt{2}xy+21\sqrt{2}=0$

Q. If the transformed equation of $xy=c^2$ when the axis are rotated through an angle of $\frac{\pi }{4}$ (in the anti clockwise direction), is $p{X}^2+q{Y}^2=r{c}^2$, then

1. $p=1$
2. $p=-1$
3. $q=1$
4. $r=2$

Q. The transformed equation of $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ when the axes are rotated through an angle of ${90}^{\circ }$ is

1. $b{X}^2-2hXY+a{Y}^2+2fX-2gY+c=0$
2. $b{X}^2+2hXY+a{Y}^2+2fX+2gY+c=0$
3. $b{X}^2-2hXY+a{Y}^2-2fX+2gY+c=0$
4. $b{X}^2-2hXY+a{Y}^2-2fX-2gY+c=0$

Q. The axes being inclined at an angle $\omega$, find the centre and radius of the circle $x^2+2xy\cos \omega -2gx-2fy=0$.

Q.

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

1. True

2. False

Q.

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

1. True

2. 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)

1. $a^2+b^2=p^2+q^2$

2. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}+\frac{1}{q^2}$

3. $a^2+b^2=p^2+q^2$

4. $\frac{1}{a^2}+\frac{1}{p^2}=\frac{1}{b^2}+\frac{1}{q^2}$

Q. The locus of the mid point of the intercept of the line $x\cos \alpha +y\sin \alpha =p$ between the coordinate axes is

1. $x^{-2}+y^{-2}=4p^{-2}$
2. $x^2+y^2=4p^{-2}$
3. $x^{-2}+y^{-2}=p^{-2}$
4. $x^2+y^2=p^{-2}$

Q. lf the axes are rotated through an angle ${60}^o$, then the transformed equation of $x^2+y^2=25$ is

1. $X^2+Y^2=1$
2. $X^2+Y^2=9$
3. $X^2+Y^2=16$
4. $X^2+Y^2=25$

Q.

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

__

Q. The transformed equation of $3x^2+3y^2+2xy=2$ when the coordinate axes are rotated through an angle ${45}^{\circ }$ is

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