Chapter 9.5, Problem 9WE

To determine

To calculate: To graph the hyperbola and show its asymptotes by dashed lines and to find the coordinates of foci

Answer to Problem 9WE

Coordinates of foci are $\left(5\sqrt{7},0\right)$ and $\left(-5\sqrt{7},0\right)$

Explanation of Solution

Given information: Equation of hyperbola is $75x^2-100y^2=7500$

Formula Used:

The hyperbola with center $\left(0,0\right)$ and foci $\left(-c,0\right)$ and $\left(c,0\right)$ and difference of focal radii $2a$ has the equation:

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ , where $b^2=c^2-a^2$

Equation of the asymptotes are $y=\frac{b}{a}x$ and $y=-\frac{b}{a}x$

The hyperbola with center $\left(0,0\right)$ and foci $\left(0,-c\right)$ and $\left(0,c\right)$ and difference of focal radii $2a$ has the equation:

  $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ , where $b^2=c^2-a^2$

Equation of the asymptotes are $y=\frac{a}{b}x$ and $y=-\frac{a}{b}x$

Calculation:

Equation of hyperbola is given as follows:

  $75x^2-100y^2=7500$

Rewriting and divided each sides of the equation by $7500$ :

  $\begin{array}{l}\frac{x^2}{100}-\frac{y^2}{75}=1\\ \frac{x^2}{{10}^2}-\frac{y^2}{{\left(5\sqrt{3}\right)}^2}=1\end{array}$

Comparing the equation with standard form of the equation of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ :

  $\begin{array}{l}a=10\\ b=5\sqrt{3}\end{array}$

Foci is calculated as follows:

  $\begin{array}{l}{b}^2=c^2-a^2\\ c^2=a^2+b^2\\ c^2={10}^2+{\left(5\sqrt{3}\right)}^2\\ c^2=100+75\\ c^2=175\\ c=\pm 5\sqrt{7}\end{array}$

Thus, Foci are $\left(5\sqrt{7},0\right)$ and $\left(-5\sqrt{7},0\right)$

Also, asymptotes are calculated as follows:

  $y=\frac{a}{b}x$ and $y=-\frac{a}{b}x$

Substituting the values:

  $y=\frac{5\sqrt{3}}{10}x$   and  $y=-\frac{5\sqrt{3}}{10}x$

  $y=\frac{\sqrt{3}}{2}x$   and  $y=-\frac{\sqrt{3}}{2}x$

Graph the hyperbola as follows:

  

Conclusion:

Hence, graph of hyperbola is sketched and coordinates of foci are $\left(5\sqrt{7},0\right)$ and $\left(-5\sqrt{7},0\right)$