Question
The angle of rotation of axes in order to eliminate xy term of the equation 
is
Hint:
Find the rotation of the equation and then find the angle.
The correct answer is:
Given That:
>>> Let the angle of rotation of axes be ϕ and let the new coordinates be x′ and y′.
>>> Thus, the old coordinates are:-
x = x′cosϕ−y′sinϕ
y = x′sinϕ+y′cosϕ
>>> Now, we will put the value of coordinates in the given equation
We know from the trigonometric identities that
⇒cos2ϕ−sin2ϕ=cos2ϕ
and2cosϕsinϕ=sin2ϕ
>>> On further simplifying the terms, we get:
>>> ⇒tan2ϕ=
>>> Therefore, the value of ϕis:-
⇒
2ϕ=
⇒ ϕ=
>>> Hence, the required angle of rotation of axes is
The Angle of rotation becomes
Related Questions to study
If the axes are rotated through an angle 45° in the anti-clockwise direction, the coordinates of 
in the new system are
The point (
) becomes (2,0) after rotation through 45 degrees in anti clockwise direction.
If the axes are rotated through an angle 45° in the anti-clockwise direction, the coordinates of in the new system are
The point () becomes (2,0) after rotation through 45 degrees in anti clockwise direction.
If the axes are rotated through an angle of 45° and the point p has new co-ordinates 
then original coordinates of p are
>>> Therefore, the original point is (3,1).
If the axes are rotated through an angle of 45° and the point p has new co-ordinates then original coordinates of p are
>>> Therefore, the original point is (3,1).
Assertion (A) : If the area of triangle formed by (0,0),(-1,2),(1,2) is 2 square units. Then the area triangle on shifting the origin to a point (2,3) is 2 units.
Reason (R):By the change of axes area does not change
Choose the correct answer
Both assertion and reason are correct and the reason is correct explanation of assertion.
Assertion (A) : If the area of triangle formed by (0,0),(-1,2),(1,2) is 2 square units. Then the area triangle on shifting the origin to a point (2,3) is 2 units.
Reason (R):By the change of axes area does not change
Choose the correct answer
Both assertion and reason are correct and the reason is correct explanation of assertion.
The point (4,1) undergoes the following successively
i) reflection about the line y=x
ii) translation through a distance 2 unit along the positive direction of y-axis. The final position of the point is
Therefore, the required point is (1,6).
The point (4,1) undergoes the following successively
i) reflection about the line y=x
ii) translation through a distance 2 unit along the positive direction of y-axis. The final position of the point is
Therefore, the required point is (1,6).
Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. On the basis of the above information answer the following The given lines will be perpendicular if
We define lines using cosine ratios of the line. While working with three-dimensional geometry (used in so many applications such as game designing), it is needed to express the importance of the line present in 3-D space. Here we were asked to find the condition for perpendicular line, so it is 
.
Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. On the basis of the above information answer the following The given lines will be perpendicular if
We define lines using cosine ratios of the line. While working with three-dimensional geometry (used in so many applications such as game designing), it is needed to express the importance of the line present in 3-D space. Here we were asked to find the condition for perpendicular line, so it is .
is equal to
Calculus and mathematical analysis depend on limits, which are also used to determine integrals, derivatives, and continuity. A function with a value that approaches the input is said to have a limit. So the answer is 1/2 for the given expression.
is equal to
Calculus and mathematical analysis depend on limits, which are also used to determine integrals, derivatives, and continuity. A function with a value that approaches the input is said to have a limit. So the answer is 1/2 for the given expression.
The value of 
is
The value of is
Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. On the basis of the above information answer the following For f=g=h=1 both lines satisfy the relation
All the above options are correct.
Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. On the basis of the above information answer the following For f=g=h=1 both lines satisfy the relation
All the above options are correct.
The lines 
and lie in a same plane. Based on this information answer the following. Equation of the plane containing both lines
Therefore, the equation of the line becomes x+6y-5z=10.
The lines and lie in a same plane. Based on this information answer the following. Equation of the plane containing both lines
Therefore, the equation of the line becomes x+6y-5z=10.
lie in a same plane. Based on this information answer the following. Point of intersection of these lines is
and 
represent :
