Chapter 9.6, Problem 20WE

To determine

To calculate:

The equation of the hyperbola by using the following foci and the different focal radii:

  $\left(-1,1\right),\left(1,-1\right);2$

Answer to Problem 20WE

The equation of the hyperbola is $x^2-y^2+2x+2y=1$ .

Explanation of Solution

Given information:

  $\begin{array}{l}\left(h,k\right)=\left(-1,1\right)\\ \left(a,b\right)=\left(1,-1\right)\end{array}$

Formula used:

The equation of the hyperbola is expressed by the following expression:

  $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$

Calculation:

The hyperbola foci are given as $\left(-1,1\right)$ and $\left(1,-1\right)$ . The difference of the focal radii is equal to $2a$ .

Now, the hyperbola is written as:

  $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$

Put the given data as:

  $\begin{array}{l}\frac{{\left(x+1\right)}^2}{{\left(1\right)}^2}-\frac{{\left(y-1\right)}^2}{{\left(-1\right)}^2}=1\\ x^2+2x+1^2-\left(y^2-2y{+}^2\right)=1^2\\ x^2+2x+1-y^2+2y-1=1\\ x^2+2x-y^2+2y=1\\ x^2-y^2+2x+2y=1\end{array}$

Therefore, the hyperbola with foci and difference of the focal radii is $x^2-y^2+2x+2y=1$ .