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 = ![\theta=\frac{1}{2}tan^{-1}[\frac{2h}{a-b}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-6444c3c62c37c18ffaeb2a18aeed4152_l3.png "Rendered by QuickLaTeX.com")
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:
⇒ ![\theta=\frac{1}{2}tan^{-1}[\frac{2(-\sqrt3)}{9-3}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-e6f2d5b1f5391cc7bef5939bb5c9ff66_l3.png "Rendered by QuickLaTeX.com")
⇒ θ = ![\theta=-\frac{1}{2}tan^{-1}[\frac{1}{\sqrt3}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-1275adbf97c13c24812d4073d3b47536_l3.png "Rendered by QuickLaTeX.com")
⇒ 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:
⇒ ![\theta=\frac{1}{2}tan^{-1}[\frac{2(-\sqrt3)}{9-7}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-44e1e206726b598a5bdf2f9fc4c6e0e1_l3.png "Rendered by QuickLaTeX.com")
⇒ θ = ![\theta=-\frac{1}{2}tan^{-1}[\sqrt3]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-e97c6dfe6673a8818a4ab6a9a28529b2_l3.png "Rendered by QuickLaTeX.com")
⇒ 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:
⇒ ![\theta=\frac{1}{2}tan^{-1}[\frac{2(\sqrt3)}{4-2}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-f10f9b5e643eb5350747788f8897395b_l3.png "Rendered by QuickLaTeX.com")
⇒ θ = ![\theta=\frac{1}{2}tan^{-1}[\sqrt3]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-6bc7effa0507d1e4fbe8df088655cac7_l3.png "Rendered by QuickLaTeX.com")
⇒ 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:
⇒ ![\theta=\frac{1}{2}tan^{-1}[\frac{2(\sqrt3)}{10-8}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-9305039b5e3a4e4619cede78d688c71d_l3.png "Rendered by QuickLaTeX.com")
⇒ ![θ = \frac{1}{2}tan^{-1}[\sqrt3]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-d5daea0d6ad554470dc61cd319f6e5e9_l3.png "Rendered by QuickLaTeX.com")
⇒ 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:
⇒ ![\theta=\frac{1}{2}tan^{-1}[\frac{2(\sqrt3)}{12-10}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-059359828dac8e8aee86321e3524d81d_l3.png "Rendered by QuickLaTeX.com")
⇒ θ = ![\frac{1}{2}tan^{-1}[\sqrt3]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-f4cd7d8f0828d054d808192bbd807bcc_l3.png "Rendered by QuickLaTeX.com")
⇒ 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:
⇒ ![\theta=\frac{1}{2}tan^{-1}[\frac{2(\sqrt3)}{8-6}]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-481d965618dd1d068f6ff78a383a326d_l3.png "Rendered by QuickLaTeX.com")
⇒
⇒ θ = ½ × π/3
⇒ θ = π/6
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