Chapter A5.3, Problem 24E

To determine

To identify: the graph of the equation.

Answer to Problem 24E

The graph of the equation is a hyperbola

  $x^{\prime 2}-y^{\prime 2}-2\sqrt{2}x+2=0$

Explanation of Solution

Given:

  $xy-x-y+1=0$

Calculation:

Given the following equation:

  $xy-x-y+1=0$

Comparing with the standard equation for a Quadratic Curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

We get:

  $A=0,B=1,C=0$

For the cross-product term to vanish, the coordinate axes must be rotated by an angle alpha,such that

  $\cot 2\alpha =\frac{A-C}{B}$

  $=\frac{0-0}{1}$

  $=0$

Therefore,

  $2\alpha =\frac{\pi }{2}$

  $\Rightarrow \alpha =\frac{\pi }{4}$

After rotation of coordinate axes, the original coordinates, $x$ and yare replaced by newcoordinates, $x^{\prime }$ and $y^{\prime }$ , such that

  $\begin{array}{l}x=x^{\prime }\cos \alpha -y^{\prime }\sin \alpha \\ =\frac{\sqrt{2}}{2}{x}^{\prime }-\frac{\sqrt{2}}{2}{y}^{\prime }\\ y=x^{\prime }\sin \alpha +y^{\prime }\cos \alpha \\ =\frac{\sqrt{2}}{2}{x}^{\prime }+\frac{\sqrt{2}}{2}{y}^{\prime }\end{array}$

Substituting the above values of $x$ and $y$ in the given Quadratic Equation,

  $\left(\frac{\sqrt{2}}{2}{x}^{\prime }-\frac{\sqrt{2}}{2}{y}^{\prime }\right)\left(\frac{\sqrt{2}}{2}{x}^{\prime }+\frac{\sqrt{2}}{2}{y}^{\prime }\right)-\left(\frac{\sqrt{2}}{2}{x}^{\prime }+\frac{\sqrt{2}}{2}{y}^{\prime }\right)-\left(\frac{\sqrt{2}}{2}{x}^{\prime }-\frac{\sqrt{2}}{2}{y}^{\prime }\right)+1=0$

  $\left(\frac{1}{2}{x}^{\prime 2}-\frac{1}{2}{x}^{\prime }{y}^{\prime }+\frac{1}{2}{x}^{\prime }{y}^{\prime }-\frac{1}{2}{y}^{\prime 2}\right)-\left(\frac{\sqrt{2}}{2}{x}^{\prime }+\frac{\sqrt{2}}{2}{y}^{\prime }\right)-\left(\frac{\sqrt{2}}{2}{x}^{\prime }-\frac{\sqrt{2}}{2}{y}^{\prime }\right)+1=0$

  $\Rightarrow \frac{1}{2}{x}^{\prime 2}-\frac{1}{2}{y}^{\prime 2}-\sqrt{2}{x}^{\prime }+1=0$

  $\Rightarrow x^{\prime 2}-y^{\prime 2}-2\sqrt{2}x+2=0$

Which is a hyperbola

Conclusion:

The graph of the equation is a hyperbola

  $x^{\prime 2}-y^{\prime 2}-2\sqrt{2}x+2=0$