Find the angle of rotation of axes to remove xy term in the equation 9x2 − 2√3xy + 3y2 = 0
Last Updated :
16 Feb, 2022
When a plane intersects a cone, conic sections, also known as conics, are created. The geometry of these sections is determined by the angle at which they cross. As a result, conic sections are divided into four categories: circle, ellipse, parabola, and hyperbola. Each of these forms has its own set of mathematical features and equations.
Angle of Rotation
The angle of rotation is a measurement in mathematics of the amount, or angle, that a figure is rotated around a given point, usually the centre of a circle. A clockwise rotation is regarded as a negatives motion, hence a 310° (counterclockwise) rotation is also known as a –50° rotation (because 310° + 50° = 360°, a full rotation (turn)). A reverse rotation of more than a total turn is typically measured modulo 360°, which means that 360° is deducted as many times as feasible until a non-negative measurement less than 360° is obtained.
Angle of rotation =
Find the angle of rotation of axes to remove xy term in the equation 9x2 − 2√3xy + 3y2=0.
Solution:
Given equation: 9x2 − 2√3xy + 3y2 = 0
Comparing this equation with ax2 + 2hxy + by2 = 0, we have:
a = 9, h = −√3, b = 3
We know, angle of rotation =
Substituting the values in this formula, we have:
⇒
⇒ θ =
⇒ θ = −½ × π/6
⇒ θ = −π/12 or 5π/12
Similar Problems
Question 1. Find the angle of rotation of axes to remove xy term in the equation 9x2 − 2√3xy + 7y2 = 0.
Solution:
Given equation: 9x2 − 2√3xy + 7y2=0
Comparing this equation with ax2 + 2hxy + by2 = 0, we have:
a = 9, h = −√3, b = 7
We know, angle of rotation =
Substituting the values in this formula, we have:
⇒
⇒ θ =
⇒ θ = −½ × π/3
⇒ θ = −π/6 or 11π/6
Question 2. Find the angle of rotation of axes to remove xy term in the equation 4x2 + 2√3xy + 2y2 = 0.
Solution:
Given equation: 4x2 + 2√3xy + 2y2 = 0
Comparing this equation with ax2 + 2hxy + by2 = 0, we have:
a = 4, h = √3, b = 2
We know, angle of rotation =
Substituting the values in this formula, we have:
⇒
⇒ θ =
⇒ θ = ½ × π/3
⇒ θ = π/6
Question 3. Find the angle of rotation of axes to remove xy term in the equation 10x2 + 2√3xy + 8y2 = 0.
Solution:
Given equation: 10x2 + 2√3xy + 8y2 = 0
Comparing this equation with ax2 + 2hxy + by2 = 0, we have:
a = 10, h = √3, b = 8
We know, angle of rotation =
Substituting the values in this formula, we have:
⇒
⇒
⇒ θ = ½ × π/3
⇒ θ = π/6
Question 4. Find the angle of rotation of axes to remove xy term in the equation 12x2 + 2√3xy + 10y2 = 0.
Solution:
Given equation: 12x2 + 2√3xy + 10y2 = 0
Comparing this equation with ax2 + 2hxy + by2 = 0, we have:
a = 12, h = √3, b = 10
We know, angle of rotation =
Substituting the values in this formula, we have:
⇒
⇒ θ =
⇒ θ = ½ × π/3
⇒ θ = π/6
Question 5. Find the angle of rotation of axes to remove xy term in the equation 8x2 + 2√3xy + 6y2 = 0.
Solution:
Given equation: 8x2 + 2√3xy + 6y2 = 0
Comparing this equation with ax2 + 2hxy + by2 = 0, we have:
a = 8, h = √3, b = 6
We know, angle of rotation =
Substituting the values in this formula, we have:
⇒
⇒
⇒ θ = ½ × π/3
⇒ θ = π/6
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