Chapter 10.5, Problem 24AYU

To determine

The appropriate rotation formula to use so that new equation contains no $xy$ term: $3x²-10xy+3y²-3=0$.

Answer to Problem 24AYU

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

Explanation of Solution

Given:

$3x²-10xy+3y²-3=0$

Formula used:

To transform $Ax²+Bxy+Cy²+Dx+Ey+F=0,B\ne 0$ into an equation in $x\text{'}$ and $y\text{'}$ 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, $3x²-10xy+3y²-3=0$.

$A=3, B=-10 \text{and} C=3$;

$\text{Cot}\left(2\theta \right)=\frac{A-C}{B}$; $\frac{3-3}{-10}=0$; $\theta =\frac{\pi }{4}$

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

$x=x^{\prime }\text{cos} 45-y^{\prime }\text{sin} 45$

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

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

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

$y=x^{\prime }\text{sin}45+y^{\prime }\text{cos}45$

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

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