Transform the equation xy=-1 by rotating the axes through…

Transform the equation $xy=-1$ by rotating the axes through the angle $\varphi=-\dfrac{\pi}{4}$.

Explanation of Method

Rotating the axes by −π/4 turns xy = −1 into a hyperbola in canonical form.

Step 1: Substitution

The old coordinates in terms of the new ones: \[x=X\cos\varphi-Y\sin\varphi=\frac{X+Y}{\sqrt2},\qquad y=X\sin\varphi+Y\cos\varphi=\frac{-X+Y}{\sqrt2}.\]

\[ xy=\frac{(Y+X)(Y-X)}{2}=\frac{Y^2-X^2}{2}=-1. \]

\[X^2-Y^2=2\quad\Longleftrightarrow\quad\frac{X^2}{2}-\frac{Y^2}{2}=1\]

Step 2: Elements

This is an equilateral hyperbola: $a=b=\sqrt2$, $c=2$, $e=\sqrt2$, asymptotes $Y=\pm X$ (the old coordinate axes). The vertices $X=\pm\sqrt2,\ Y=0$ are the old points $(1,-1)$ and $(-1,1)$. The foci are $(\sqrt2,-\sqrt2)$ and $(-\sqrt2,\sqrt2)$.

$X$ $Y$ $(1,-1)$ $(-1,1)$ $-\tfrac{\pi}{4}$

$xy=-1$ in the axes rotated by $-\pi/4$: $X^2-Y^2=2$. The vertices are $(1,-1)$ and $(-1,1)$, on the new $X$-axis.

Result

\[\boxed{\,X^2-Y^2=2\quad\Longleftrightarrow\quad\frac{X^2}{2}-\frac{Y^2}{2}=1\,}\]