Chapter 10.5, Problem 10AYU

To determine

Whether, to eliminate the $xy$-term from the equation $x²-2xy+y²-2x+3y+5=0$, rotate the axes through an angle $\theta$, where $\text{cot}\theta =B²-4AC$ is true or false.

Answer to Problem 10AYU

False.

Explanation of Solution

Given:

$x²-2xy+y²-2x+3y+5=0$,

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 $\text{Cot}\left(2\theta \right)=\frac{A-C}{B}$.

Calculation:

Consider the given equation $x^2-2xy+y^2-2x+3y+5=0$,

$A=1;B=-2;C=1$

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 $\text{Cot}\left(2\theta \right)=\frac{A-C}{B}$.

Hence, $\text{Cot}\left(2\theta \right)=B²-4AC$ is false.