Rotation of Axes

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.

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 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 $2a^2+3b^2$

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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. 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. 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 $x^2+6xy+8y^2=10$ 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 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. 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. $b{X}^2-2hXY+a{Y}^2-2fX-2gY+c=0$