Chapter 9.5, Problem 34WE

To determine

To calculate:

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

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

Answer to Problem 34WE

The equation of the hyperbola is $x^2-y^2-2ax+2ay=a^2$ .

Explanation of Solution

Given information:

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

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 $\left(a,a\right)$ and $\left(-a,-a\right)$ . 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:

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

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