Rotation of Axes

Q. If a straight line through $C(-\sqrt{8},\sqrt{8})$ making an angle ${135}^{\circ }$ with the x-axis cuts the circle $x=5cos\theta ,y=5sin\theta$ in points A and B, then length of segment AB is

1. 5
2. 10
3. 15
4. 

Q.

If $p$is the length of the perpendicular from the origin to the line, whose intercepts with the coordinate axes are $\frac{1}{3}$ and $\frac{1}{4}$ then the value of $p$ is

1. $\frac{3}{5}$

2. $\frac{1}{12}$

3. $\frac{1}{5}$

4. $5$

Q.

Chord of contact of the point (3, 2) w.r.t. the circle $x^2+y^2=25$ meets the coordinate axes in A and B. The

circumcentre of triangle OAB is

1. 

2. 

3. 

4. DOES NOT EXIST

Q.

The origin of the coordinate axes is shifted to (3, 4) and the axes are rotated through an angle of ${30}^{\circ }$ in the clockwise direction. Find the new coordinates of (7, -2)

1. 

2. 

3. 

4. 

Q.

If the axes are shifted to the point (1, -2) without rotation. What do the equation $2x^2+y^2-4x+4y=0$ becomes?

Q.

Intersecting the x-axis at a distance of 3 units to the left of origin with slope -2.

Q.

Co-ordinate axes are rotated through an angle of ${45}^{\circ }$ in the anti-Clockwise direction. Find the Co-Ordinates of (2, 3) in new co-ordinate system.

1. 

2. 

3. 

4. 

Q. find the equation of the parabola with vertex (2, -3) and directrix x-4y-48=0

Q.

Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinates axes.

Q.

Find equation of the line through the point (0, 2) making an angle $\frac{2\pi }{3}$ with the positive x - axis. Also, find the equation of line parallel to it and crossing the y - axis at a distance of 2 units below the origin.

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.

Line $L$has intercepts $a\&b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line $L$ has intercepts $p\&q$, then

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

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

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

4. none of these

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.

If P = (1, 0), Q = (-1, 0) and R = (2, 0) are three given points, then the locus of a point S satisfying the relation

$S{Q}^2+S{R}^2=2S{P}^2$ is

1. A circle with centre at the origin

2. A straight line parallel to y-axis

3. A straight line parallel to x-axis

4. A circle through origin

Q.

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. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of $5$
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0, 0)

Q.

If P = (1, 0), Q = (-1, 0) and R = (2, 0) are three given points, then the locus of a point S satisfying the relation

$S{Q}^2+S{R}^2=2S{P}^2$ is

1. A circle through origin

2. A circle with centre at the origin

3. A straight line parallel to y-axis

4. A straight line parallel to x-axis

Q. A point is on the $x$-axis. What are its $y$-coordinate and $z$-coordinates?

Q.

The co-ordinate axes are rotated through an angle of ${60}^{\circ }$ anti-clockwise. Find the vector corresponding to 1+i in new co-ordinate axes.

1. (1+i)

2. (1+i)

3. (1+i)

4. (1+i)

Q. Fill in the blanks:
(i) The $x$-axis and $y$-axis taken together determine a plane known as______
(ii) The coordinates of points in the $XY$-plane are of the form_____
(iii) Coordinate planes divide the space into_____ octants

Q. 51. Find the equation of the line equidistant from the lines 9x+6y-7=0 and 3x+2y+6=0.

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. -1

2. 0

3. 1

4. none of these

Q.

$\begin{array}{cc}Linepassesthroughthepoints& Slopeoftheline\\ p.(1,6)and(-4,2)& 1.0\\ q.(5,9)and(2,9)& 2.-3\\ r.(-2,-1)and(-3,2)& 3.\frac{4}{5}\\ s.(4,0)and(3,3)& 4.\frac{5}{3}\end{array}$

1. p - 3 q - 1 r - 2 s - 4

2. p - 1 q - 3 r - 1 s - 2

3. p - 1 q - 3 r - 2 s - 2

4. p - 3 q - 1 r - 2 s - 2

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.

A circle is drawn touching both the axes. The equation of a chord with P(3, 2) as midpoint is x=3. If P lies one unit away from the centre of the circle, find the length of the chord.

1. √5

2. √3

3. 2√3

4. 2√5

Q.

Find the coordinatesof the foce, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$16x^2+y^2=16$

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 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.

For any three sets A, B, C, prove that :

(i) AX(B intersection C) = (AXB) intersection (AXC)

Q. 79.Find the equation of the parabola whose focus is (a, b) and whose directrix is x/a+y/b=1.