Chapter 10: Q 26. (page 677)

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$11 x^{2} + 10 \sqrt{3} x y + y^{2} - 4 = 0$

Short Answer

Expert verified

$x = \frac{\sqrt{3} x^{'}}{2} - \frac{y^{'}}{2} y = \frac{x^{'}}{2} + \frac{\sqrt{3} y^{'}}{2}$

Step by step solution

Step 1. Given information

An equation is $11 x^{2} + 10 \sqrt{3} x y + y^{2} - 4 = 0$ .

Step 2. Determine the appropriate rotation formulas.

The given conic is given by the equation given below.

$11 x^{2} + 10 \sqrt{3} x y + y^{2} - 4 = 0$

The rotation formulae for the given conic such that $x y -$ term is absent are given by the formulae given below, where $θ$ can be determined by the relation $\cot 2 θ = \frac{A - C}{B}$

$x = x^{'} \cos θ - y^{'} \sin θ y = x^{'} \sin θ + y^{'} \cos θ$

The value of the $θ$ can be determined by substituting the values of $A, B, C$ in the expression that evaluates the value of angle $θ$

$\begin{array}{rcl} \cot 2 θ & = & \frac{11 - 1}{10 \sqrt{3}} \\ = & \frac{π}{6} \end{array}$

Therefore the rotation formulae for the given equation of conic are given by the equations that follow.

$\begin{array}{rcl} x & = & x^{'} \cos \frac{π}{6} - y^{'} \sin \frac{π}{6} \\ = & \frac{\sqrt{3} x^{'}}{2} - \frac{y^{'}}{2} \\ y & = & x^{'} \sin \frac{π}{6} + y^{'} \cos \frac{π}{6} \\ = & \frac{x^{'}}{2} + \frac{\sqrt{3} y^{'}}{2} \end{array}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$11 x^{2} + 10 \sqrt{3} x y + y^{2} - 4 = 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the appropriate rotation formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Applied Mathematics

Statistics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no xy-term.11x2+103xy+y2-4=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the appropriate rotation formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Applied Mathematics

Statistics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$11 x^{2} + 10 \sqrt{3} x y + y^{2} - 4 = 0$