Chapter 10.5, Problem 26AYU

To determine

The appropriate rotation formula to use so that new equation contains no $xy$ term: $11x^2+10\surd 3xy+y^2-4=0$.

Answer to Problem 26AYU

$x=\frac{1}{2}\left(\surd 3x^{\prime }-y^{\prime }\right)$, $y=\frac{1}{2}\left(x^{\prime }+\surd 3y^{\prime }\right)$

Explanation of Solution

Given:

$11x^2+10\surd 3xy+y^2-4=0$

Formula used:

To transform $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0, B\ne 0$ into an equation in $x^{\prime }$ and $y^{\prime }$ without an $x^{\prime }{y}^{\prime }$-term, rotate the axes through an angle $\theta$ that satisfies the equation $\text{Cot}\left(2\theta \right)=\frac{A-C}{B}$.

$x = x^{\prime }\text{cos}\theta - y^{\prime }\text{sin}\theta ; y= x^{\prime }\text{sin}\theta + y^{\prime }\text{cos}\theta$

Calculation:

Consider the given equation $11x^2+10\surd 3xy+y^2-4=0$.

$A=11, B=10\sqrt{3} \text{and} C=1$;

$\text{Cot}\left(2\theta \right)=\frac{A-C}{B}=\frac{11-1}{10\sqrt{3}}=\frac{10}{-10\sqrt{3}}=\frac{1}{\sqrt{3}};2\theta =\pi /3;\theta =\pi /6$

$x = x^{\prime }\text{cos}\theta - y^{\prime }\text{sin}\theta$

$x = x^{\prime }\text{cos}30 - y^{\prime }\text{sin}30$

$x=x^{\prime }\left(\frac{\surd 3}{2}\right)-y^{\prime }\left(\frac{1}{2}\right)$

$x=\frac{1}{2}\left(\surd 3x^{\prime }-y^{\prime }\right)$

$y = x^{\prime }\text{sin}\theta + y^{\prime }\text{cos}\theta$

$y = x^{\prime }\text{sin}30 + y^{\prime }\text{cos}30$

$y=x^{\prime }\left(\frac{1}{2}\right)+y^{\prime }\left(\frac{\surd 3}{2}\right)$

$y=\frac{1}{2}\left(x^{\prime }+\surd 3y^{\prime }\right)$